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eriAl TO) OPO 
WU TY fri, Fy 


L161—O-1096 


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7 
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Longmans’ Modern Matbematical Series 


INFINITESIMAL 
CALCU S 


SECON LOIVGEL 


BY 


Fe ey Ara ola aie ONi bpaly 


PROFESSOR IN THE UNIVERSITY OF LIVERPOOL 


WITH DIAGRAMS 


Te CSINSC EVER Fog. Cx. bx Ey Foe Ni cA ND GG, 
20go bk Anika N Otol B RR OW ..1 ON DL OON 
FOURTH AVENUE & 30TH STREET, NEW YORK 
BOMBAY, CALCUTTA AND MADRAS 
1917 


All rights reserved 


Wedricl< 


43 Oct 1% 


a 


Mathe tie 
matics 4 og Sevgnens wt VAR “\3 MAA%& 
N 


le Jhcdel le JM Ods) 


Tus book is divided into two sections: the first deals with those 
parts of the Infinitesimal Calculus which have been recently intro- 
duced into the syllabus of some examinations for higher school 
certificates, while the two sections taken together correspond fairly 
closely to the curriculum of students reading for the first part of 
an honours course in mathematics or for the ordinary degree in 
arts, science or engineering. 

Believing that there is no royal road which leads smoothly and 
directly to the Infinitesimal Calculus, the author has made no 
attempt to evade all the difficulties which at the outset face the 
student in this subject. The road has, however, been laid in the first 
section so as to pass through those domains of number and function 
with which the student is probably already acquainted, while the 
functions which are likely to be unfamiliar to him have been reserved 
for the second section. 

To assist the student in mastering the fundamental conception 
of a differential coefficient, two ideas which are usually reserved 
for books of a more advanced character have been introduced 
at the beginning and used throughout the book, namely, range and 
sequence, and the ordinary symbolism in connection with them 
has been varied. The symbols to express open and closed ranges 
have, however, been changed so slightly that the alteration amounts 
to little more than a typographical modification; but a more 
important change has been made in the arrow notation, which is 
now so generally used to express the limits of a sequence. The 
author is indebted to his former colleague, Dr. James Mercer, for 
the beautiful suggestion of the arrow with a single barb, either 
upper or lower. This notation has been successfully tested in class 
work, but has not previously appeared in print. Teachers who 
have recognised the great advantages which have resulted from the 


OH) 
SCIAP IOU 


vi PREFACE 


introduction of the fully barbed arrow may perhaps be willing to 
try the experiment of arrows with a single barb. 

No attempt has been made in the first section to deal with the 
definite integral, nor has the usual notation for the indefinite integral 
been introduced until a comparatively advanced stage. It need 
hardly be said that it is no part of the author’s plan to exclude such 
an important and universally adopted symbol; his plea for the 
postponement of its use is based upon the impossibility of 
justifying the symbol {o(x)dw as a representation of inverse differ- 
entiation until the nature of a definite integral has been explained. 
Something also may be gained by introducing students at an early 
stage to a differential equation, even though it is of the simple 


type ‘ty, 
AB me (x) 

The book is not written for any particular group of students ; 
it is designed for those who wish to use the Infinitesimal Calculus 
as an instrument in the attainment of further knowledge. It is 
therefore essentially a book of practical mathematics. With this 
end in view, fundamental ideas are explained at great length ; for 
the easiest and quickest way to master this subject is to acquire a 
firm grasp of the conceptions upon which it is based. The student 
is advised to return again and again to the earlier chapters ; it is 
only gradually that the matter contained in them can be assimilated, 
and the knowledge which the student acquires as he progresses 
will sometimes furnish the key to open some doors that may be 
closed to him at a first attempt. The Infinitesimal Calculus cannot 
be learnt ‘either by memory or by mimicry—it needs judgment and 
reflection ; but if studied in the right spirit it may, by strengthening 
these qualities of mind, fulfil one of the most valuable aims of 
education. 

Historical notes and references have not been given. The student 
who is interested in this side of the subject should turn to Dr. 
Whitehead’s Introduction to Mathematics, and to Mr. Jourdain’s 
Nature of Mathematics ; these valuable books are readily accessible, 
and will naturally conduct the student to Mr. Rouse Ball’s more 
systematic writings on the history of the subject. In these various 
books he will find the natural complement and the fitting reward 
of the serious work which the study of a text-book implies. 


PREFACE mas 


The author acknowledges most gratefully the help of many 
friends. Dr. Macaulay, an editor of the series in which the book 
appears, has given practical assistance in reading the manuscript 
and in following the book through the press; he has also given 
a generous encouragement, which has been as invaluable as his 
criticism. Dr. Proudman has read the earlier chapters in manu- 
script more than once, and proofs of the whole; there are many 
parts of the book which have assumed their final shape under his 
skilful direction. Mr. R. Hargreaves and Mr. R. O. Street have 
read the proofs, and a deep debt is owed to them for many valuable 
criticisms and useful suggestions. 

The exercises are numerous, as they must be in a book on 
this subject. They afford examples for the student both on his 
first reading and also when he returns to the subject and goes over 
it a second or a third time. The author will welcome corrections 
of errors. In cases where it seems desirable that the attention of 
the solver should not be distracted by the form of the answers, 
these have been placed at the end of the book. 


Beer ek Cee 


GO ING EONG IES 


SEI OSEIHOUNO EE 


El DeGe TUF 9 


NUMBER, FUNCTION, GRAPH 


ARTICLE 


Numbers - - - - - : 2 & F s 
Functions - - - - - a 2 r : fe 
Polynomial functions - - - i zy 3 4 
Notation of functions - - 2 é = a 4 
Graphs of mathematical functions - - - : - 
Other functions - - - - : : 5 i f 
A power-function - - - : : 2 : 3 
Graph of the linear function - - “ : * : 
Graph of the quadratic function . . - - - 
The arithmetical continuum - - 2 3 E : 
Range of a variable - - - : " : 
Natural range of the independent variable of a function - 
Relative magnitude of two functions - - - - - 
Illustrative examples - - - 2 : : 
Exercises I - - - a . ‘ E 2 


(SaVaded Mele ai 


LIMIT, CONTINUITY 


Irrational numbers - - - - - - - - 
Illustrations of a limit - . - - - - - 
Conditions for a limit of a sequence - - - - - 
Example of a sequence with a rational limit - - - 
Monotone and other sequences - - - s - - 
Notation for a limit - - - - - - - - 
Discussion of f(x) in the neighbourhood of the valuex=a - 
Evaluation of f(z) when x = a, an irrational number - - 
Discussion of f(z) in the neighbourhood of a value of x at 

which the function is undefined” - - - - - 


x CONTENTS 
ARTICLE : 
24. Continuity and discontinuity of f(r) atv=a . - - 
25. Summary with illustrations - - - : - % 
26. Limit of the sum, product and quotient of two functions, 
whenz—>a - - - - - - . - F 
27. Continuous functions . - - - : J : 5 
28. Examples to illustrate limits” - - : r e : 
Exercises II - : - - : : - é 
a Bis ie A Ras aE 
DIFFERENTIAL COEFFICIENT 
29. Definition - - : - - = : i 3 2 
30. Illustrative examples : - - “ : : i 
Exercises IIT (a) - - - : ° 3 i : 
31. Remarks upon the definition of a differential coefficient - 
32. Notation - - - - : i : 2 
33. Geometrical illustration - - 8 ! M . . 
34. Second notation - - - . 2 S : . 
35. Rules for differentiating the sum, product and quotient of 
two functions whose differential coefficients are known - 
36. Illustrative examples - - - - - - 3 
37. Differential coefficient of x", when n is integral - - - 
38. The differential coefficients of sin x, cos x, tan x- - - 
39. The differential coefficients of cot x, sec x, cosec x - - 
40. Examples on differentiation - - - : = 2 
Exercises III (B) - : - - ¢ i f 2 
Cre Sau od ane  ! 
THE SIGN OF THE DIFFERENTIAL COEFFICIENT 
41. Derived function - - - = - : - . 
42. Meaning of the sign of /’(x) - - - - - Fi 
43. Stationary values of f(z) - - - - - : 
44. Graphical interpretation of the sign i S’ (x) - - - 
45. Maximum and minimum values - - - - - 
46. Relation between the graphs of f(z) and f(x) - - . 
47. Examples of maxima and minima - - - - - 
Exercises IV__ - - - - : 2 - a i 
Oe Ba EB Le AY 
ALGEBRAIC FUNCTIONS 
48. Quadratic functions - - - - . = = : 
49. Cubic functions - - - - - - = u “ 


60 
61 


CONTENTS > 

ARTICLE ; PAGE 
50. The infinities of linear, quadratic and cubic functions - - 63 
51. Illustrative examples - - - - - - - 64 
52. Rational algebraic functions - . - - - - 65 
53. Reciprocal of the polynomial function - = Se - 66 
54. The general type R(x) = P,,,(x)/P,,(x) - - - - - 67 
55. Illustrative examples - : - - - - - 68 
56. The function given by y?= R(z) - he - - 70 
57. Illustrative examples - - - - - - - 71 
Exercises V - - - - - - - - - 72 


CL Eels he aaa val 
INVERSE OF A FUNCTION, FUNCTION OF A FUNCTION ~ 


58. The square root function - - - - - - - 75 
59. General inverse functions - - - - - - - 76 
60. Some properties of inverse functions - - - - - 76 
61. The graph of an inverse function - - - - - 76 
62. Differential coefficient of an inverse function = - - - 77 
63. The differential coefficient of 4/x - - - - - 78 
64. The differential coefficient of x/2 - - - - . 78 
65. Function of a function, compound function . - - 79 
66. The differential coefficient of f[o(x)] - - - - - 80 
67. Illustrative examples - - - - - - - 81 

Exercises VI (A) Inverse functions’ - - - - - 82 

Exercises VI (B) Differentiation - - - - - 82 


Gre tO EL 
TANGENT AND NORMAL 


68. Equations of the tangent and normal of y= f(z) : - 84 
69. Illustrative examples - - - - - - - 84 
70. The algebraic equation of acurve~ - - “ - - 85 
71. Illustrative examples - - - - 5 - - 86 
72. Subtangent and subnormal - : - - - - 87 
73. Parametric representation of a curve - - - - 87 
74. Value of dy/dx, when x and y are functions of a parameter 88 
75. Equations of tangent and normal to the curve 2= /f,(é), 
= fa) ae Nh dasa sh Uy of oe wh eaanalle MB 
Exercises VII - - - - - - - - - 89 


GHAR RRA NHI 


SECOND DIFFERENTIAL COEFFICIENT 


76. Definitions - - - . - - - 92 
77. Symbol D for Titerea nation : - - ~ - : 93 


pam as laa ial i a alate ceil ie nent een 1 EE ti 
ig: 3 


xii CONTENTS 
ARTICLE ; 
78. Geometrical meaning of the sign of the second differential 


79. 
80. 
81. 
82. 
83. 
84. 
85. 
86. 
87. 
88.. 
89. 


107. 
108. 


coefficient - - - ~ : wile & 
Relation between the graphs of f(x), f’(w) and f’(x)_ - 
Criteria for maxima and minima - - - 
Illustrations of f’(x) - - - : . : , 
A parabolic approximation to y= f(x) . - - 
The circle of curvature - - - “wee : 
Curvature - - - - : Z : Z E 
Formulae for curvature’ - - - - - : 
Illustrative examples - - : : ‘ : 
The orders of magnitudes - - - : : 2 
Velocity and acceleration - - - - - : 
Leibniz’s theorem - - = : : : 


Exercises VIII (A) Second differential coefficients, points 


of inflexion - = : * 2 = “ é 
Exercises VIII (B) Radius of curvature - ~~ : 
Exercises VIII (c) Motion in a straight line - - 


Gm Bay ad OK OBS 
INVERSE DIFFERENTIATION 


General remarks - - - - - - - 
Definition of inverse differentiation - - - - 
Geometrical meaning of inverse differentiation - - 
Notation of an inverse function - . : - 
Notation of inverse differentiation - - - : 
Integral of x”, provided #-1_ - - - - 
Integral of a sum of two functions - - - - 
Integral of the product of a constant and a function 
Integration of a polynomial’ - - - - - 
Integration of a general polynomial - - - - 
Integration of a function of a function of x - - 
The logarithmic function - - - - - - 
Differential coefficient of log,x - - - - 
Integration of the reciprocal of the linear function - 
Integration of certain cases of R(x) - - - - 
Illustrative examples - - - - - - 
The value of D-1!z—1, when = is negative - - - 
Integration of sin x, cosa, tan x, sec x, sin? x, cos*x - 
Exercises IX - - - : - - 


CAC FS BRITS | 6 
AREAS, VOLUMES 


First application of integration - : - 7 


Differential coefficient of a trapezoidal area A(z) ~ 


108 
108 
109 
109 
110 
111 
111 
111 
111 
112 
112 
113 
114 
115 
116 
117 
117 
118 
119 


121 
122 


ARTICLE 
109. 
110. 
hl. 
112. 
113. 
114. 
115. 
116. 
ee 
118. 
119. 
120. 
121. 


122. 
123. 
124. 
125. 
126. 
127. 


128. 
129. 


130. 
Pol. 


CONTENTS 


Area bounded by a curve, the axes and an ordinate - - 
Integraph - - - - - - - - - 
Illustrative examples - - - - - - - 
Extended meaning of A(x) - - - - - - 
New interpretation of the relation between f(x) and f’(a) 
Volumes of solids of revolution - - - - - 
Illustrative examples - - - : - - 
Approximative equation of a curve given by three points - 
Simpson’s rule for a trapezoidal area - - - - 
Extension of Simpson’s rule - - - - - - 
Simpson’s rule for volumes of revolution - - - - 
Trapezoidal areas whose bases rest upon Oy - - - 
Areas of certain oval curves and the volumes generated by 

their revolution - - - - - - - - 
Exercises X (A) Areas” - - - - - - - 
Exercises X (B) Volumes - - - - - - - 
Exercises X (C) Simpson’s rules - - - - - 


(Ol BAP HG Dake 2G 
MOMENTS BY INTEGRATION 


The moments of a system of particles distributed along Ox 
First and second moments of a line-distribution of matter 
Illustrative examples - - . - - - - 
First and second moments of a lamina - - - - 
Illustrative examples - - - - - - - 
Second, or axial, moment of a lamina about an axis 

perpendicular to its plane - - - - - - 
Second, or axial, moments of volumes about an axis of 

symmetry - - - - = - - - - 
Illustrative examples - - . - - - - 
Centroid or centre of gravity - - - - - - 
Illustrative example - - - - - - - 
Exercises XI _ - - - - - - - - - 


ANSWERS 


xii 
PAGE 
122 
123 
123 
125 
126 
127 
128 
129 
130 
131 
131 
131 


132 
133 
134 
135 


137 
137 
138 
139 
141 


141 


142 
143 
143 
144 
144 


Oy ee een ge 


A vere yea haas 
oc exeevsr hth ob 
* os Cae 


ce 


Vee e 


i = 
Mis 


SVXOII COIN TE 


Arla ed SIS vl 


NUMBER, FUNCTION, GRAPH 


1. Numbers. 

The numbers which are used in Arithmetic and Algebra are often 
qualified by adjectives ; thus, we speak of positive and of negative 
numbers, of integral and of fractional numbers. By each of such 
qualifications the existence of a class of numbers is suggested. 
For if we say that 2 is an integer, we imply (i) that there are 
other integers, such as 3, 4, 5, ..., and (ii) that there are numbers, 
such as 4, 3, ..., which are not integral. 

Numbers are divided first of all into positive and negative 
numbers, zero not belonging to either class. To every positive 
number there corresponds a negative number such that the addi- 
tion of these two numbers is zero, for example, 

2+(-2) =0 

In the further classification of numbers which we make here, we 
shall consider only the class of positive numbers, remembering 
that to each positive number there corresponds a negative number. 

Integers claim our attention first. They may be written in the 
following natural order yb Ry rae 


Such a set is the type of a sequence, or progression, of numbers, 
the distinguishing property of a sequence of numbers being that it 
is always possible to write down the number which occupies the 
mth place. An example of such a sequence with which the student 
is familiar is given by 10uSi Gude 
in which the nth place is occupied by the number 12 — 2n; this 
sequence is an arithmetical progression. Other familiar instances 
of sequences of integers are 

De re or ed L 

Per AE ANd DERE AEC DRE 

Pas Wie: Be 3 eee (Gm Ltt 


C.C, A 


2 INFINITESIMAL CALCULUS CHAP. 1 


After integers the class of fractions has to be taken. Fractions 
are divided into various sub-classes ; we have reduced and non- 
reduced fractions ; some fractions also are proper, while others are 
spoken of as improper. But, whatever sub-classes are introduced, 
all fractions are included in the sequence 

$ hts BES BEET BEEBE H-- 

In this sequence the reader will notice that } and 2/4 both occur. 
It is true that these are equal, but there is a sense in which they 
may be distinguished ; for 4 implies the division of the unit into two 
parts, whereas 2/4 implies that it is divided into four parts. Again, 
the inclusion of such numbers as 3/3 and 4/2 may seem to outstrip 
the ordinary interpretation of fractions. Those who feel doubt 
about the inclusion of such numbers in the above catalogue may 
strike them out and be assured that the reduced list contains all the 
numbers which they regard as fractions. The sequence is divided 
into blocks by semi-colons, the successive blocks containing every 
fraction the sum of whose numerator and denominator is equal to 
3, 4, 5, 6, 7,.... Thus, in the 20th block the fractions would be 


1 2 3 2 


BU) 80) 103 bee 
In the sequence it is possible to place any given fraction ; thus, 
if 19/21 is considered, the sum of its numerator and denominator 
is 40; this fact places it in the 38th block ; its actual place in the 
complete sequence is found to be 722. The fractions in each block 
of the sequence are arranged in order of magnitude ; in the sequence, 
as a whole, this is not the case. It is, in fact, clearly impossible to 
arrange fractions i in order of magnitude : for, between any two given 
positive fractions, a/b and c/d, the fraction (a + c)/(b + d) lies. 


Positive integers and fractions can be derived from unity by a 
finite number of two of the fundamental operations of Arithmetic, 
addition and division. Thus we have as illustrations 
l+1+1+1+1 
Ne et rR ns 
at uleeaTeeT 


I 


| Go Ot 


But there are numbers which cannot be obtained by a finite 
number of these operations ; some such numbers are familiar to 
the reader, e.9., 

PLO G EI LTO ort Made Alera yd), 
These are called irrational or incommensurable numbers ; they will 
be discussed below. 


2. Functions. 
The simplest illustration of a function is provided by a mathe- 
matical table, which, however it is arranged, consists essentially 


, 


CHAP. I FUNCTION 3 


of two columns, or rows, of numbers placed so that the numbers 
which are opposite to each other correspond. Ordinary mathe- 
matical tables contain information about squares, square roots, 
cubes, cube roots, logarithms, antilogarithms, sines, cosines, tangents, 
log sines, log cosines, and log tangents. In the case of each of 
these tables there is a process which allows us to verify that the 
number on the right hand corresponds to the number on the left 
when the table is arranged in column ; moreover, the same pro- 
cess which gives the numbers in the table allows of its extension to 
include pairs of corresponding numbers which are not given in 
the table. If we look first at a simple case, the table of squares, 
and arrange the table horizontally, we have as an abbreviated 
specimen of a table 


Here x and x? are variables, that is, symbols which stand for any 
number from a given class. This table might be extended at any 
point ; thus between 1 and 2 we might insert nine numbers by 
taking x to every tenth, 


L21 | 144 1-69 1-96 2-25 2-56 | 2-89 3°24 | 3-61 


In general, a correspondence between two classes of numbers in which 
to each number of the first class corresponds a number of the second class 
is called a functional relation. Again, the variable which corresponds 
to numbers in the first class is called the independent variable, and 
that which corresponds to those in the second class is the dependent 
variable. In this way we may say that there is a functional relation 
between the independent and dependent variables, or, as it is more 
usually stated, the dependent variable is a functzon of the indepen- 
dent variable. 

In the first section of this book the reader’s attention is specially 
directed to single-valued functions in which the dependent variable 
is derived by simple algebraical operations from the independent 
variable. 


All the functions mentioned above are single-valued, that is, 
there is a strict one-one correspondence between the independent 
and dependent variables. An exception may suggest itself in the 
case of the square root ; in its most general form, corresponding 
to any admissible value of ~ we have two values of the square root, 


4 INFINITESIMAL CALCULUS CHAP. I 


namely, + +/x; but it is usual in the table to give only + 4/z, to 
which the name of principal value is attached. 

The number of the entries in a table is necessarily restricted ; 
but there are two other kinds of restriction in constructing a table 
of a function which may be noted, the first of these results from 
the nature of the function, the second is imposed by practical utility. 
Thus, the square roots of negative numbers cannot be given ; 
negative values of x in the table for »/a do not therefore occur ; 
but the squares of negative numbers are not given in the table 
for x*, because they can be deduced immediately from the results 
given for positive values of wz. Still another restriction of a 
different kind occurs in the case of such a function as 


a* — 2 
x+1 


The table of this function is, to give a few values, 


but no value can be given to the function when a = — 1. In this 
function a single isolated value of x occurs for which there is no 
corresponding value of the dependent variable. At a later stage 
this exception will be considered. 

The tables of trigonometrical functions are not extended beyond 
the first quadrant ; extension being unnecessary, as simple formulae 
exist which enable us to calculate the values of the ratios of angles 
in the second, third and fourth quadrants when we know those in 
the first. The formulae in the case of the sine and tangent 
function are 


sin (90° + x) = sin (90° — 2) sin (180° + x) = —- sina 
sin (270° + x) = — sin (90° — 2) 


tan (90° +x) = — tan (90° — x) tan (180° + x) = tanz 
tan (270° + x) = — tan (90° — zx) 


The use of 10 as a base of logarithms allows us to print in a small 
compass a large amount of information, as in this system the log- 
arithms of positive numbers with the same digits all have the same 
mantissa. There is in the logarithm table also a natural restriction, 
as log x has no arithmetical value when 2 is zero or negative. 

A further feature of mathematical tables is that they are usually 
calculated to 4, 5, 7 places of decimals, or in rare cases to 12 or 15 


\ 


CHAP, I FUNCTION 5 


places ; the character of the work determines the kind of table 
which is most suitable. But it is worthy of note that in tables of 
surds, logarithms and trigonometrical functions the numbers 
corresponding to rational values of x are in general irrational ; no 
table can, therefore, give more than approximate values, unless 
in very exceptional cases; the only instances of rational results 
in the tables of trigonometrical functions are afforded by sin 30’, 
cos 60°, tan 45°, and when the angle is zero or 90°. This does 
not imply that there are not angles whose _ trigonometrical 
functions are rational; these must occur, but the ratio of such 
an angle to a right angle is certainly incommensurable, and will 
not be included in tables calculated on the ordinary basis of angular 
measurement. 

The word function is often used in cases in which no mathematical 
process can be laid down for establishing a correspondence between 
two classes of numbers given by observation or experiment. But 
when the observations are sufficient to justify the formulation of a 
functional relation underlying the two classes of numbers, the 
observer is justified in regarding his observations as governed by a 
law. Such a correspondence is known to physicists to exist between 
the numbers denoting the pressure and the volume of a gas at 
constant temperature and within certain ranges of pressure. The 
particular function which expresses this relation in its mathe- 
matical form is known as Boyle’s Law. 


_ 3. Polynomial functions. 

It is not possible to make an exhaustive classification of functions 
such as we attempt in the case of numbers. But certain functions 
can be conveniently grouped together. The first group we consider 
is that of rational algebraic functions, in which the mathematical 
process by which the function is derived consists of a finite number 
of the four fundamental operations—addition, subtraction, multipli- 
cation and division—performed upon x and upon certain constant 
numbers, a, b,.... 

Thus the linear function is defined as the result of multiplying x 
by a and adding 6, where a and b areconstant. The linear function 
implies a correspondence between two classes of numbers which ~ 
are typically written f 
ax +b 

As instances of linear functions we take 

2a x -—l - 3x - 4 4a + 2 

Again, if we multiply a linear function by x and add a constant c, 

we obtain a(ax + 6) +e = ar? + bx +0 


the typical form of the quadratic function. 


6 INFINITESIMAL CALCULUS CHAP. I 


Proceeding in the same way, we obtain from the quadratic function 
the cubic function 


x(ax? + ba +¢) +d = ax? + bx*? + cr +d 
These functions are instances of the rational integral algebraic 
SUNCLION/ OLY, i pan a paved veatos ae Lee he belly 


in which a, b, c, ... k are the constant coefficients of the 7 ,+ 1 terms, 
k being called the constant term. This is briefly described as the 
polynomial function of the nth degree. 

Another class of functions is obtained by taking the quotient 
of two polynomials ; such a quotient is called a rational algebraic 
function of x. 


4, Notation of functions. 

Just as it is convenient to have a single symbol x which stands 
for any number of a given class, so it is convenient to have a sym- 
bol which expresses any one of the functions described above, or 
indeed any function which comes under the general definition. 
The symbol used is f(x) ; it implies some rule involving mathe- 
matical operations with numbers which allows us, when a value of 
« is assigned, to give the corresponding value of f(x).* Reverting 
to the definition of a single-valued function, the reader will see 
that the two corresponding classes of numbers which constitute the 
function are expressed now by the pair of symbols 


xv 
f(x) 


a correspondence which in its expanded form may include such 
a table as 


Tae Ltd Oe lala ae 


f(- 2) re 1) | ©) | fay | fQ) 


The second class of numbers is expressed by the symbol y; with 
this notation the correspondence which constitutes the functional 


relation is written x 


y 
Both notations are used contintially. The second possesses an 
apparent defect, inasmuch as the correspondence between a par- 
ticular pair of values is not indicated in the notation; it has, 
however, the advantage of indicating the two corresponding classes 
each by a single letter. There are devices which allow us to remove 
*The reader may compare the symbols of sin (a), tan(x), log (x) with f(#). The 


brackets round a, which are usually omitted in writing sin 2, tana, log a, are retained 
in f(z). 


CHAP. I FUNCTION, GRAPH 7 


the apparent defect just indicated ; one is the device of attaching 
the same numerical suffix to corresponding w’s and y’s. Thus the 
extended table of the general function can be written 


Ya Yn 


y 1 | ¥2 | Ys 
where the suffix indicates the correspondence between the pairs of 
mated variables. These marks justify no inference as to numerical 
magnitude, x, may be greater or less than xw,; they are simply 
distinguishing marks similar to ma., mi., tert., by which boys in a 
school who have the same surname are sometimes distinguished. 
A similar notation of dashes is also used, but in this subject the 
dashes are reserved for another purpose. 

The union of the two notations just explained leads to the relation 


y = f(a) 

It is perhaps convenient to remind the reader that of the two 
classes of numbers the primary class is that which consists of the 
values of the independent variable x, and the secondary class that 
which consists of the dependent variable y. 


5. Graphs of mathematical functions. 

The preceding brief sketch of a function will have suggested that, 
at any rate, certain functions may be represented by means of 
curves. The notation of x, y for the expression of a function has 
reminded the reader of Cartesian coordinates. In the system 
devised by Descartes every point on a plane is represented by two 
numbers which express the position of the point relative to two 
(rectangular) axes in a plane, the determining numbers being 
denoted by x (the abscissa) and y (the ordinate). If now we think 
of a pair of corresponding values of the variables x, y connected by a 
functional relation and also mark on a paper the point whose 
position is determined by these values considered as coordinates, 
we have a method of representing each corresponding pair of values 
of the two classes of numbers in the table of a function by means 
of a point. The table on p. 3 can now be extended by writing in 
a third row the letters corresponding to the points determined by 
the numbers above it. Thus the new form of the table for the 
square-function is written as 


1 
AWGN Cu hi: 


8 INFINITESIMAL CALCULUS OHAP. 1 


In this table we indicate that A is the point whose coordinates 
are (— 2,4), B is (— 1, 1),C is-(1, 1), D is (2,4) and # 1s (3, 9). 
Thus, upon the squared paper we have, see Fig. 1, six points which | 
correspond to the six values of the function given in the table. 
If the number of pairs of values of the function be increased by 
taking, for instance, a sequence of values of x increasing by tenths, 
we obtain between the successive pairs of points nine new points. 
In this way the student will probably have no difficulty in con- 


Bara ees 
ACE 


BIGi i: 


vincing himself that the graph which corresponds to a square- 
function is a curve. The curve is called a parabola, and in the 
language of analytical geometry its equation is said to be 


y= 2 


The same fact is often expressed by saying that the parabola is 
the graph of the function 2?. 


The student is expected to know that the graphs of linear functions 
are straight lines, and that those of quadratic functions are parabolas 
which have their axes parallel to Oy; the different lines which 
correspond to the different linear functions obtained by changing 
the constants a and 6 in ax + b and the various parabolas which 
correspond to the functions az? + bx + c will be discussed below. 


The graph of a function presents to us a rough epitome of the 
chief properties of the function : thus, a sine-curve suggests periodi- 
city, while the parabola is the type of a curve with either a single 
maximum or a single minimum. The table of the sine-function 
extends only from 0° to 90° and cannot suggest any property except 
that of a function continuously increasing over its range of tabula- 
tion; the table of squares again does not present the principal 


mien FUNCTION, GRAPH 9 


property of the function 2”, because it is only constructed for positive 
values of a On the other hand, the rough graph of the sine- 
function cannot take the place of the table of sines even when 
the table is constructed only for each minute of angle. The truth 
is that the graph and the table play different parts in moulding our 
conception of a function. The discussions in this book require 
us to consider the table in its widest extent as the fundamental 
basis of the function, but invaluable help is often derived intuitively 
from the graph. By itself the table of a function suggests that 
the function consists of a number of pairs of values, a very large 
number, but still a limited number ; the graph corrects this view 
and suggests that the number of corresponding pairs of values is 
unlimited. The graph does more, for it implies the converse, namely 
that to each point on the curve there corresponds a pair of values 
of x and y. 


6. Other functions. 


Most of the graphs with which the student has to do consist of 
smooth or regular curves which can be traced through a certain 
number of guiding points by means of a flexible ruler. But there 
are functions which are incapable of representation in this way. 

Suppose that we consider the price at different times of a loaf of 
bread of given weight. Here time is the abscissa or independent 
variable (x), and corresponding to each value of x there is a value of 
the dependent variable, an ordinate (y), which is the price of the 
loaf of given weight. But as the price can only be changed by a 
multiple of the smallest coin of the realm (or, at any rate, by a 
sub-multiple of it), the graph must consist of finite straight lines 
parallel to the axis of w, the length of each line representing the 
interval during which the price remains unchanged. To draw a 
smooth curve through two points whose ordinates correspond to 
3d. and 34d. would imply that the price of a loaf could be and was 
md., a sum which cannot be liquidated satisfactorily in any existing 
monetary scheme. 

An example of a non-mathematical function which can be repre- 
sented by a smooth graph is afforded by the relation between the 
weight (y) of a man and his age (x). For, in this case, although 
our scales cannot record such a weight as 41/5 stone, yet we know 
that if a man ever scales 9 stone there must have been a time at 
which he weighed 44/5 stone. A smooth curve drawn through a 
certain number of points whose coordinates are (x, y) would give a 
satisfactory representation of the man’s weight. 


7. A power-function. 

The graph of 2, see Fig. 2, is of importance in itself and because 
it enforces the distinctions between different classes of numbers 
which were made at the beginning of the chapter. First, we consider 


10 INFINITESIMAL CALCULUS CHAP. I 


positive integral values of « and obtain a sequence of isolated 
points on the graph whose coordinates are 


(1,2) (2,4) (3,8) (4,16) (5, 32)... 


Secondly, we take a sequence of negative integral values of x 
and obtain 


(-1,4 (— 2,4 (-3, 4) (- 4, x's) (— 5, 345)... 

A gap between the two sequences of points is filled by defining 
2° as equal to 1. The next step is to give wa fractional value m/n, 
where m and m are positive integers, and to define 2” as (2/2)™. 
_ Thus, we may interpolate nine values between 0 and 1 from the 
following table 


Bel ou itpoldtug! 4o-ae tt oeel noe. loc Mates | 0-9 


2° | 1:07 | 1-15 | 1-23 | 1-32.) 1-41 | 1-62; 1:62 1-74 | 1:87 


It would be possible to give a great many other pairs of values, 
but the results given will probably indicate sufficiently well the 
course of a smooth curve which passes through the given points. 


Fre. 2. 


But it would be an error to suppose that the complete curve was 
given by taking only fractional values of x; this consideration 
takes no account of such a point as 


(2, 2?) 
which certainly forms part of the complete graph. It may perhaps 


be added that the meaning assigned to 2°, when 2 is irrational, is 
based upon the fact that we seek to render the complete graph of 


CHAP. I FUNCTION, GRAPH 11 


2* a smooth curve ; the student may note that the various extensions 
of the meaning of 2* as x became negative, zero and fractional, 
result in the production of a graph which has the characteristic 
of being a smooth curve. 


8. Graph of the linear function. 
_ Writing f(x) = ax + 6, it is obvious that 


JO=6 fyH=aidb f2) =2a+0... 
This sequence of values indicates that the points which correspond 
to x = 0, 1, 2, ... ie upon a straight line whose gradient is a. This 
graph is studied so fully in books on Algebra that it is sufficient 
to remark that, whatever values x and h may have, 
fi +h) — f(x) . -afv +h) + 6 — (ax + 6) a 
h i h = 

In the diagram, Fig. 8, 

MP =f@) NQ=faw+h RQ=faw +h -f@) RQ/PR=a 
It is only in the linear function that this simple 
result holds. 

If we fix P and take a point Q’ on PQ, or PQ 
produced, we have, by similar triangles, R’ being 
the intersection of PR and the ordinate of Q’, 

Te TD 
Tae Pa 
It follows that Q’ is on the graph, and 


therefore that every point of PQ and of PQ 
produced in either direction is situated upon the graph of az + 6. 


Fic. 3. 


9, Graph of the quadratic function. 

We write f(x) = az? + 2bx + ¢, the coefficient of x being taken 
for convenience as 2b; this does not imply that the coefficient is 
an even number, as 0 is not necessarily an integer. The function 
will be discussed in a later chapter with other methods of analysis, 
but it is useful to resolve it now into the sum or difference of two 
squares. Thus, 


f(z) = ax? + 2ba2 +c 
eae c b b? — ac 
= a(x? + 2+) =a(2+2) - 


a 


tay 
Be ee) 


the alternative forms being used, according as 6? > or < ac. 


12 INFINITESIMAL CALCULUS CHAP. I 


The graph of f(x) is a parabola whose axis of symmetry is parallel 
to Oy. The following features of the curve are deduced from the - 
algebraic form in which it is written. 

i. The parabola is concave to an observer at a great distance 
above Ox, if ais positive ; convex, if a is negative. 

ii. The equation of the axis of symmetry is x = — 6/a. 

iii. The crest or hollow of the curve is at a distance (ac — b?)/a 
above the axis of 2. 

iv. If b? < ac, the parabola does not cut the axis of x ; if b? = ac, 
it touches Ox; if b2 > ac, it crosses the axis of x at two points 
which correspond to the values of x which make f(x) = 0, that is, 
at points whose abscissae are 

2 
Z b re 4/(b? — ac) 


a a 


It may help in drawing the parabola to notice that all quadratic 
functions in which a has the same numerical value, regardless of 
sign, can be drawn with the same parabolic ruler or template. 
Thus, the graphs of 

x — (x — l)? v2 +] 4 — 3? 


are copies of each other ; they differ in their positions relative to 
the axes. 


B' 


Fic. 4. 


It is worth noticing, too, that all parabolas are similar, just as 
all circles are similar. In the diagram, Fig. 4, the parabolas 


y= 2 y = 102? 
are drawn. The equation of the second curve may be written 
(10y) 2=(10z)4 


CHAP. I NUMBER 13 


a form which shows that, if the curve y = 10a? is magnified, so that 
the scales of x and of y are both increased in the ratio of 1: 10, the 
magnified curve coincides with the parabola whose equation is 
y = x*, as drawn on the original scale. In the diagram OAB is the 
curve y = 10z?, OA’B’ is the same curve drawn upon a magnified 
scale. 


10. The arithmetical continuum. 


The totality of all real numbers, however they may be classified, 
constitutes the arithmetical continuum. To form an adequate 
notion of this most important conception is difficult; it may 
help the student in his attempt to realise it to suggest that 
he should, in the first instance, concentrate attention upon that 
part of the continuum which lies between 0 and 1 and reflect 
upon the non-terminating decimal 


0. abcdef... 


where each of the letters a, 6, c, ... stands for one of the integers 0, 
1, 2, 3, 4, 5, 6, 7, 8, 9. The totality of the numbers thus repre- 
sented constitutes the portion of the continuum between 0 and 1. 
If in the general number written down above, all the digits after 
a certain point are zeroes, the number is a rational fraction whose 
denominator is of the form 275%, where p and q are integers ; 
other rational fractions are expressed by means of decimals which, 
after some point, recur. But, besides the class of rational frac- 
tions which is said to be countable because it can be arranged in 
a sequence, there is the larger class of incommensurable or irra- 
tional numbers which is uncountable, because it cannot be arranged 
in a sequence. 


We may also briefly consider the geometrical representation of a 
number. On the axis of « we take the points O and 4, the first 
of which represents zero and the second represents unity. There 
are elementary geometrical methods by which a point P lying 
between O and A, may be determined so that the ratio of OP to OA 
will represent any proper rational fraction. Every irrational 
number can also be represented by a point ; this statement, which 
cannot be proved, is known as the Cantor-Dedekind postulate ; 
its formulation constituted a valuable step in mathematical 
progress. To illustrate the general method of representing an 
incommensurable number, 44/2 is taken, for which a simple direct 
construction is also available. The first step in the general pro- 
cedure is to divide OA into tenths and to take OP,/OA = 0-7, 
and then to divide each tenth into one hundred parts and take 
P,P, as seven of these parts, so that OP,/OA = 0-707, and further 
to take P,P,/OA = 0-0001, which gives OP,/OA =0-7071. The 
process is as unending as the evaluation of +/2 as a decimal 


14 INFINITESIMAL CALCULUS CHAP. I 


fraction. The Cantor-Dedekind postulate asserts that there is 
a point P on the axis such that 


OP/OA = }4/2 


The student to whom the process of working with sequences is 
new will find additional information in Chapter II. The construc- 
tion of P can be effected directly by drawing a square whose side 
is 430A and marking off upon Ox a line OP equal to the diagonal 
of this square. The reader will perhaps realise better the necessity 
of the Cantor-Dedekind postulate by attempting the representation 
of such a number as 4, for which no exact geometrical construction 
involving a finite number of operations is available. 

This process of construction for the continuum may be carried 
out between every consecutive pair of integers, positive or negative ; 
in this way an idea of each section of the continuum may be formed. 
But there is one other question which must be considered before 
leaving this slight sketch, namely, What are the extremities of the 
continuum ? An answer of a definite nature is impossible, for the 
continuum must contain every number which we may require to 
use. The extent of the continuum is therefore indefinite. An 
appropriate symbol to express this fact is given in the next article. 


11. Range of a variable. 


If all the numbers of the arithmetical continuum from z = a to 
x = b are taken, they represent a range of values of the variable. 
Now this range is called a closed range, if the end-valuesx = a, x = B, 
are included ; it is then written (a, b). But, if the end-value x = a 
is excluded while x = 6 is included, it is called a range open at 
the left end, and will be denoted by [a, b) ; if x = ais included, but 
x = bis excluded, it is a range open at the right end, and is denoted 
by (a, 6]; a range open at both ends is called an open range, and 
is denoted by [a, 0]. 

With this notation the arithmetical continuum is denoted by 
[- «©, ], the positive part of the continuum by [0, « ] and the 
negative part by [-— , 0], zero being a number which is neither 
positive nor negative. The symbol © may be defined for our 
present purposes by the statement that [0, « | includes every posi- 
tive number, however large it may be. 


12. Natural range of the independent variable of a function. 

The notation explained in Art. 11 allows us to describe succinctly 
the values of the variable for which a function is defined. Thus, 
the polynomial (or rational integral algebraic) function of a is 
defined for every value of x which lies in the. range [—- 0, o ], 
that is, it can be calculated for any given value of x. The re- 
ciprocal function 1/x is defined for all values in the continuum, 
except x = 0; its range therefore consists of two parts, namely, 


CHAP. I FUNCTION 15 


[ — «, 0] and (0, © ], for from both of these ranges x = 0 is excluded. 
The logarithmic function log x is defined for a range of x denoted 
by [0, « ], that is, for any positive value of x Such a function 
as 4/(a2 — «?) is defined for values of 2 in the range (- a, a), the 
function being zero at the end-values of the range. 


13. Relative magnitude of two functions. 


There are six ways of expressing the relative magnitude of two 
functions, namely, 


J(x) > o(x) (greater than) 
II. f(x) < (x) (less than) 
III. f(x) } (x) (not greater than) 
IV. f(x) < (x) (not less than) 
V. f(x) = o(x) (greater than or equal to) 
VI. f(x) = o(@) (less than or equal to) 


Of these III and VI are identical, and II differs from them only 
in values of x for which the functions are equal; again, IV and V 
are identical and differ from I only for those values, if any, at which 
equality subsists. 


Each of these six relations suggests an enquiry, namely, the range of 
values of x for which the inequality holds. In answering such queries 
the notation of the previous article of closed and open ranges provides 
an exact method of expression which the student should practise. It 
is often advisable to draw the graphs of the two functions; it may 
also be of help to determine by analysis the points of intersection of 
their graphs. The examples given below illustrate the two methods 
and their combination. 


14. Illustrative examples. 
Ex. 1. To find the range of values of x for which 


= 20 — 3 > ae + 2 
We see from their graphs that 


y =-2xr- 3 y= 34 + 2 
cross at the point (- 1, - 1); also from this point in the direction 
Gir increasing, ¢—= 57-2 is above’ y= — 27 — 3) while son ¢the 
other side of the crossing point y= - 2% - 3 is above y= 32 + 2. 


It follows that Rowe gy gis '5 


for values of x less than - 1; the interval is open at both ends, (i) 
because infinity is counted as an open end, and (li) because at x= - 1 
the functions are equal. The required range is [- o, - l]. 


Ex. 2. To discuss the values of x for which 


x+6 


Ae 
a x+1 


16 INFINITESIMAL CALCULUS 


It is easy to replace the problem given by substituting for it the 
inequality 


62 -1= 2 
e+ 1 
Now the graph of 6x - 1 is a line, while that of 5/(% + 1) is a rect- 
angular hyperbola, the graphs meet at « = # and at w = - 13, for 


these values of x the functions are equal. The diagram, Fig. 5, in 
which the graphs are superposed shows that the inequality holds for 


two ranges (- 14, - 1] and (%, 0]. 


ee 
- 


- 


a 
w~ 
Ks Go 
pe 


me ae ae oe 
ere wee 


ae co cakes kas coo wes ae 


Fic. 5. PIG: 


Ex. 3. Yo find the ranges of x for which 
1 a | 
e-1~ @-1P 
The two functions are equal when wz = - 2, 3. 


An examination 
of the graphs of the functions, see Fig. 6, will convince the student that 


while both functions are undefined at x= 1, the graph of 1/(x — 1) is 
above that of (a - 7)/(«- 1)? at all values of vw except x=1 in the 
open range [- 2, 3]. The inequality thus holds in [ - 2, 1} and in 
[1, 3]. The beginner who is unpractised in drawing the graphs of such 
functions as are here discussed will find information on the subject in 
Chapter V. 

As it is legitimate to multiply both sides of an inequality by a positive 
number, it might at first sight seem possible to replace the above 
inequality by el te nT 


but when x = 1, (x — 1)? is not positive. 


EXERCISES I 


1. Trace the graphs of the following linear functions of x 


i. w- 2 li, -a2- 3 iii. 3a + 1 
iv. -— 2x- 4 v. 444+ 1 vi. 4(~@ — 2) 
vii. — 3$(3x - 4) viii. - v7 - 1 ix. 102 — +4 


10 


CHAP. I 


KXERCISES 


2. Write down linear functions whose graphs pass through the pairs 
of points whose coordinates are 


CHAP. I 17 


WO} (20) rea {ey al ay firm (OS) (aaa) 
iv. (2, 4) (1, 0) ey een eel yet) Vit (oe 2G roma) 
vii. (4, 3) (3, 4) Mili (Gul eB) ee bebe arias (Laan b)( See) 


3. Trace the graphs of the following quadratic functions of x 


hed aee e ON | li, — w+ 1 iii. 4,2 
iv. 1022 yy, ee 4 9) vi. (a — 4) (a + 3) 
vii. (4 — a) (w + 3) Vill. 27 + 2e + 5 ix. 327+ 6x - 1 

x. 2u7+ 5a + 2 Brg fl Nearer daee xii. 5a? — Ta + 2-5 


4. Find quadratic functions whose graphs pass through the triads of 
points whose coordinates are 


ie 2s 1). (0).0Y-(2,01) ii. (0, 5) (1, 6) (3, 14) 
nee a i(0) 1),(10 1) Gr (irate). 101), Gtrad) 
v. (0, 0) (1, 3) (3, 15) Vinitlen © 2yt 2d 8)y Se) 


vii. (— 1, -3:1) (0, — 2) (1, 0°9). viii. (0, 2) (1, 0) (2, 6) 


5. Find the range, or ranges, of values of x for which the following 
inequalities hold 


if 


z+1l>Po ii. 


= 27 -—-5> 0 


WY ae ee ce 1Vrot abe ore ew 
Vv, o- 1 = 0 Vib tates 1 
vii. (2 —- 1)(8- x2) <0 vill. (v + I)\(v + 2)=0 
ixs2(2e—- 5) > 0 Kaeo ectisnn Ot ea G) 
x1. (% +1)(3 — 22) > 0 Sue Gee bias 10 
xiii. lla< 10 — 622 KV 6a il 1Oyz 
BV ed ie 0) XVinvoceine — 1 ie 0 
xvii. v3 < Qa? xviii. at < 223 
xix. 1+ 4/(7 +2) < 10 xx. 4/(~@ + 1) << 4/(7 - 2) 
xxi. 4/(@+ 5) - «/a@>1 pa ab Vint ees arg Ware 
xxii Dee as ot XXIV Cae el AG a 2) — 9 
e+2 “+10 x(x + 3) 
x> — 4 3 oa at pa 
XXV. fe = 2) as ~ wo ap =o OEE Sogn oes TNs 13 


6. In the sequence of rational fractions given in § 1, show that the 
place occupied by a/b is }(a + b — 2)(a + b — 3) +4. 

7. Show that all proper rational fractions may be arranged in a 
sequence. Show that in one of these sequences the place occupied by 
a/bis3(b — 1)(6 — 2) +a. 


CHAR Ta healt 
LIMIT, CONTINUITY 


15. Irrational numbers. 

We begin by examining how a physical quantity, for instance 
the length of a straight wire, is measured. Supposing that its 
length lies between 2 and 3 metres, we first mark off 2 metres from 
one end, and then consider the length left ; if this part lies between 
6 and 7 decimetres, the total length is between 2-6 and 2-7 metres. 
A further estimation of the part remaining after 2-6 metres have been 
marked off gives a length lying (say) between 2 and 3 centimetres, 
thus making the total length between 2-62 and 2-63 metres. With 
a millimetre graduation we might then get an estimate of the length 
as greater than 2-624 metres and less than 2-625 ; and with a vernier 
we might assign two other numbers 2:6247 and 2-6248 between 
which the length lies. Thus, measurement of length, and the same 
is true of all physical measurement, implies two sequences of 
numbers ; these sequences are in the case of the wire 


2, 2-6, 2-62, 2-624, 2-6247 
3, 2:7, 2-63, 2-625, 2-6248 


The power of extending these sequences depends upon the per- 
fection of the observer and his instruments. But every measurement 
involves two such sequences, and their extension is theoretically 
possible. 


The expression of an irrational, or incommensurable, number 
corresponds very closely with the above method of measuring a 
physical quantity, and in the case of the irrational number we are 
generally better able to realise the possibilities of extension of the 
sequences. Thus, the two sequences which define 7 are 


3, 31, 3:14, 3-141, 3-1415, 3-14159, 3-141592 
4, 3:2, 3-15, 3-142, 3-1416, 3-14160, 3-141593 
In the case of the length of the wire and of the value of x, 


the sequences indicate a convenient way of separating the rational 
numbers into two classes, the first of which contains those which 


CHAP. II LIMIT 19 


are less than the quantity to be defined, while the second class 
contains those which are greater. 

By a slight. modification we may indicate any number N by 
a partition of the arithmetical continuum into two classes, one 
of which contains all numbers not greater than N, while the 
second contains all that are greater. 


16. Illustrations of a limit. 


There is one idea which must be acquired, or rather developed, by 
every one who studies the infinitesimal calculus ; it is the conception 
of a limit. Before attempting to frame formal statements con- 
cerning it, the idea will be discussed in relation to two examples 
with which the reader is familiar. 


The first illustration is drawn from recurring decimals. In 
arithmetic it is stated that 2-2 and 22 are equivalent. Let us 
examine the meaning of this statement. ‘Now, 2-2 is an abbreviation 
of the non-terminating decimal 


2-2222... 
which, by the method of the previous article, can be defined by the 
two sequences 9 9.9 9.99 9.999 9.2999 .. . . (A) 
Bi Pp a PAP BH PAV VBE WAP PPB wie cg 18) 


In working with a number which is defined by sequences, we 
content ourselves with the approximation suited to the purposes 
for which the number is required; thus, if we are satisfied with 
four places of decimals, we take 2-2222. Let us estimate the 
discrepancies (or errors) which occur in our work, according as 
we take the various terms of the sequence A instead of 22; these 
errors form a sequence which in vulgar fractions is written 


o ’ v o doo D000 ee Be hE (C) 


The 10Ist term of C is 2 10-1, which is < 10-1 ; the 10Ist term 
of A differs then from 22 by a “quantity less than 10° 100 thats: 
the 101st term of A agrees with 22 for 100 decimal places. Moreover, 
if we assign a magnitude 10-", where r has any integral value, 
however large, we may assign a term in the A-sequence, namely 
the (r + 1)th, which differs from 22 by less than 10-. 

Again, the errors in excess made by taking the terms of the 
B-sequence for 22 are Ly ae Che abla aloe an bk ilby 
and the same argument shows that the (r + 1)th term of the 
B-sequence differs from 2% in excess by a quantity less than 10-”. 

It is thus immaterial, with the degree of accuracy which we 
accept, whether the (r + 1)th term of the A or of the B-sequence 
is used. From this we see that, to our degree of approximation, 
the numbers of either of these sequences, after a certain point, may 


20 INFINITESIMAL CALCULUS CHAP. II 


be replaced by 22. This fact is expressed otherwise by saying that 
25 1s the limit of the sequence A or of B. It is in this sense that 
we may say that 22292... = 22 


The above process is illustrated in Fig. 7, in which, along an axis, 
we measure OP = 2, PQ = 0-2, QR-= 0-02, ... and also measure 
OX = 2%. The terms of the sequence A are then represented by 


OP, OQ; (OR. ss 2 nn 
those of the sequence C by } 
PX, QA: RAGES Cac or 


The sequence of points P, Q, R,... lies to the left of X but approaches 
X, the points crowding together so closely that the geometrical 
treatment cannot be pursued to advantage. Now, although no 
point of the sequence coincides with X, yet if we agree to consider 


Raa re all a el Ne eS LIE 
6 PR 


PLGueve 


two lines as equal, provided the difference in their lengths is less than 
any assigned small magnitude, we may say that the lengths of the 
lines in the A’-sequence from and after some term are equal to 
OX ; for, after this term, the lengths of the corresponding lines of 
the C’-sequence are negligible according to the assigned standard. 


The second illustration of a limit is drawn from the phenomenon 
of motion, as the first has been taken from counting. We shall 
examine the meaning of such words as “ walking four miles an 
hour.” This phrase does not imply that the pedestrian has to walk 
four miles, or that he has to walk for one hour in order to walk at 
four miles an hour. Indeed, he may have walked four miles in an 
hour without ever having walked at four miles an hour. 

As a method of calculating velocity we take the plan which is 
adopted by the authorities who set traps for motorists. Two 
policemen, X and Y, are stationed at A and B respectively, at a 
measured distance apart, on a road. As the motorist passes A, 
X signals to Y, who starts his stop-watch ; as the motor passes B, 
Y presses the stop-watch again and completes his observation. 
The time ¢ seconds registered by the watch gives the time in 
which the motor traverses AB, which we take as 220 yards. On the 
assumption of uniformity, the speed works out at 

" x bl eu miles per hour 

Sie cong ca ear P 
If there are variations in speed, the maximum speed certainly 
exceeds this mean speed. The single observation is sufficient for 


CHAP. It LIMIT 21 


the authorities. But if we wish to determine the velocity at A, 
we should have to place observers at posts nearer and nearer to A ; 
let us suppose that the sequence of distances from A of these ob- 
servers measured in yards is 


Ey Or een Castes 
while the sequence of observed times in seconds is 
Nee ata 


Then a sequence of observed mean speeds measured in yards per 
second is given by Delite a 
Ay MOTE TA 

If the observers make no errors, we have here asequence of numbers 
which more and more nearly approaches the value of the velocity 
at A. If we are satisfied with an accuracy of one place of decimals, 
and find that all the terms of the sequence after a certain point agree 
to one place of decimals, we may take this value as the speed at A. 
It would be unfair to expect great accuracy from this method ; the 
best speedometer is a very rough instrument. The existence of 
the above sequence and the possibility of its extension, which is 
implied by the dots, justifies us in speaking of the speed at A. 

The speed at A is the limit of the sequence, just as 22 is the limit 


of the sequence 2 99 9.99 9.999 


To estimate speed directly is difficult ; it is generally ascertained 
by means of indirect methods ; but to understand these methods the 
process of measuring speed as the limit of a sequence must be 
mastered. 

One objection may very properly be made by the reader; he 
may ask why the last observation made is not sufficient. The 
answer to this is, that theoretically there is no last observation, for 
the sequence by which the speed is defined does not end. A second 
difficulty may arise from the fact that in the first illustration two 
sequences were used, and in the second only one. The reader may 
resolve his doubts on this point by removing the sequence B from the 
discussion of the recurring decimal, and thus satisfy himself that 
a single sequence is sufficient in defining a limit. 

A distinction can be drawn between the nature of the sequences 
which define the recurring decimal and of that by which speed is 
determined. In the first case the terms of the sequence were pre- 
scribed by the nature of the decimal approximation, while in the 
second the sequence of speeds depended upon the arbitrary disposition 
of the various observers. Really the distinction does not constitute 
an important difference ; both sequences effect a separation of the 
numbers of the continuum into two classes, and it is this separation, . 
or partition, by which the number which is the limit is defined. 


22 INFINITESIMAL CALCULUS. CHAP. II 


17. Conditions for a limit of a sequence. 
The general typical representation of a sequence is 


Q4,; Qe, As, eee Qn eee An+m> eee 


where the suffixes define the places of the terms. We wish to 
find under what circumstance we are justified in saying that this 
sequence has a definite limit, that is, that a single number is defined 
by it. 

Let us suppose that there are two persons engaged in settling 
the question ; A, who wishes to establish that the sequence defines 
a number, and B, who has to be convinced of the validity of the 
process. The first step is taken by B, who agrees to lay down a 
scale of accuracy with which he is satisfied ; he may require an 
accuracy corresponding to four places of decimals, he may be 
content with less, he may want more than four places. We shall 
suppose that he demands an accuracy corresponding to 7 places 
of decimals. It remains now for A with this requirement before 
him to determine a value of m such that the difference (without 
regard to sign) between the mth and any succeeding term is less 
than 10~-*-1, that is, he has to indicate a value of n such that 


Qnim ~ an < LOso5- 


for all values of m. If he can do this, B must acknowledge that 
he may use a,, or any succeeding term of the sequence, in work 
in which a degree of accuracy corresponding to r places of decimals 
is required. The value of a, gives the value of the limit of the 
sequence to 7 places of decimals. The difficulty of A’s task lies 
in the fact that B may vary h's demand by changing the value of r, 
and A has therefore to show that a value of m can be assigned which 
satisfies the fundamental inequality, whatever the value of 7 may be. 
If he does this, the sequence has a limit whose value to r places 
of decimals is given by ay. 
In the sequence 
11-3, 1-444, 1-3+4++4-4,.. 

which defines 47, a certain B carried his doubts so far as to assign 
to r a value over 600; A, who happened to be the same person as 


B, successfully overcame the difficulties of the problem and evaluated 
x to 616 places of decimals. 


A sequence may have a rational or an irrational number as its 
limit. In the case of a rational limit, the fraction or integer defined 
by the sequence is a simpler form of expression than the sequence, 
but if the limit is irrational, a sequence may be the only way of 
expressing the number which is the limit. In either case the 
sequence may be regarded as determining the number which is 
its limit. 


CHAP. II SEQUENCE 33 


18. Example of a sequence with a rational limit. 
Taking the sequence 1, #, f, 45, 34,... 


where a, = 2 — 2-"+1, we can show that the sequence has a limit and 
that this limit is 2. 
First, let us find the value of m which must be taken, ifr = 4. We 


have Onem — Oy = 27M — Qrn-m+1 — g-nt1(] — Q-m) 


Now, since (1 — 2~™) is a proper fraction, it suffices to find a value 
of n which makes 2-"+1 < 10-°, This is satisfied by n = 18. 


If we take the more general case in which 10~"*! is assigned, we must 


that is log;p2 tt << -— 4r 
Hence n —1>1r/log, 2 
and n> 1 4 r/log, 2 


Again, the limit of the sequence is clearly 2, because n can be taken 
so that a, ~ 2 is as small as we please. 


19. Monotone and other sequences. 


There is one class of sequences of great importance, which includes 
those mentioned above, namely, the sequences in which the terms 
either do not increase or do not diminish; such sequences are 
called monotone. As examples of such sequences, we may take 


1 1 
it Diy 4, 4> q; d, ee 


1, 2, 2, 3, aE oF eee 
The reader will be able to convince himself by studying the 
graphical representation of a sequence that a monotone sequence 
tends either to a definite limit or to infinity. The series of points 
whose coordinates are 
(1, ay), (2, A>), (3, Az) , “sie (n, An), Se 
constitutes a graphical representation of the sequence 
ee tee al ey ss, Dyed. 
Examples of monotone sequences which have an infinite limit are 
ee Os <b aera TW 
I has Ap Aley es, PA seared fie 
Non-monotone sequences which do not satisfy the test of Art. 17 
are illustrated by 
a, =2+(-1)", 6, =sindnn, c, = {1 + 3(-)2}C* 


The limits of the sequence a, are 3 and 1, of b, are 0, + sin4r, 
+ sin 27, and of c, are 4 and — 3. These limits are obvious on 
writing down a few terms of the sequences. 


24 INFINITESIMAL CALCULUS CHAP. II 


20. Notation for a limit. 


If a, is the nth term of a sequence which has a definite limit a, 


Re write lima, =a, OF d,—>a 


n being increased indefinitely. 

Again, if a, is the nth term of a monotone sequence, we write in 
the case in which the sequence is increasing (or, more correctly, 
non-decreasing) 


| A, >a 
while, if it is decreasing, we write 
a, —~a 
the single barb* of the arrow indicating increase when the under 
barb is used, and decrease when the upper barb is drawn. 


When a,, increasing or decreasing, becomes greater arithmetically 
than any magnitude that can be assigned, we say, rather paradoxi- 
cally, that the limit of a, is infinite. We distinguish (i) the case 
in which a, exceeds any positive magnitude that can be assigned, 


writing tn + © 


and (ii) the case in which a, is less than any negative magnitude, 

however large its magnitude apart from sign may be, and we write 
i ae aon 

in both cases n is infinitely large, or as we may put it in this 

notation, n +o. 

21. Discussion of f(x) in the neighbourhood of the value x=a. 


Let a,’, a,’, a’, ... be an ascending sequence whose limit is a, so 
that a,’ ~a; also let aj, dg, as, ... be a descending sequence with the 
same limit, so that a, — a. Then, if f(x) is defined for each 
number of the sequence and if 


f(ay’) ? f(az') ’ F(a), a) 
has a limit which is the same for all ascending sequences whose limit 
is a, we call this limit the left-hand limit of f(~) at x = a, and write it 


Lim/f(z), «x«—a 
Also if HOM RICA JGR 


has a limit which is the same for all descending sequences whose limit 
is a, we call it the right-hand limit, and indicate it by 


Rlim f(z), «>a 
Further, if the function is defined at x = a, we have a third 
value f(a). 


* The arrow notation may be read thus: —> ‘tends to,’—> ‘tends up to,’ — ‘tends 
down to.’ 


CHAP. II UNDEFINED VALUE 25 


Various cases arise according as the magnitudes 
Llimf(z, f(@, Rlmf(z)  (*# a) 


are equal or not. These cases will be studied and illustrated. It is 
important to notice that when f(x) is defined at x = a and in its 
neighbourhood, the above three magnitudes may exist. If f(x) is 
undefined at x = a, but is defined in its neighbourhood, we may 


then have Lliim f(z), Rlmf(x) («#—-a) 


although we cannot substitute x = a in f(z). 
The genesis of certain types of undefined values is indicated 
below in Art. 23. 


22. Evaluation of f(x) when x=a, an irrational number. 

The process described in the previous article may be illustrated 
by the procedure which is followed sometimes for evaluating f(x), 
when « is irrational. Let us take a particular value, say, x =m 
of f(x) = (a + x + 1)/(@? — x + 1). Two sequences which define 


Meng D Miotlawolss Woulaly BtlalLba!, 
Beg.) 23-157) 8142, 3-14 16573 
In the first sequence of values x —7, and in the second «~7; 
substituting in f(x), we obtain two sequences 


f(3), f(3-1), f(3-14), f(3-14]), eres 
Upon evaluation the following sequences for f(x) are found 


1857 1:826 1-8135 1-8132 1-813068 
POLop mL IGP a LS 17) Warl8 129) vil Sr30388 


23. Discussion of f(x) in the neighbourhood of a value of x at which 
the function is undefined. 


The function f(x) is undefined at x=a, when the mathematical 
processes which suffice for its determination at ordinary values fail. 

Thus, for instance, if f(x) is a quotient of the functions (2) 
and (x) which are defined for all values in (a, 6), then, provided (2) 
does not vanish for any of these values, f(x) may be calculated from 
the tables of o(”) and (2) for all values of xin (a, 6). Butif U(x) =0 
fore vallcsaOl a = 27. 75..4. 0.1m this range, then f{() cannot, be 
calculated for any of these values of x, and f(x) is undefined at 
fe tay ee Vy. LWO) Cases arise which are illustrated below ; 
in the first Ua, ) = 0, but (&,) = 0, while in on second o(z,) =*0, 
U(z,) = 0, 7 being a number i in the set Lowa 

The simplest case * of an undefined ain is afforded by the 
reciprocal function 1/x when a = 0; here it is no answer to say 


* Another instance is afforded by tanx (~-—>90°). The erroneous statement that 
tan 90° = o is so common that the student is recommended when he has mastered 


26 INFINITESIMAL CALCULUS CHAP. It 


that 1/x is 0 ,when x — 0, the true answer being that as x — 0, 
1/x ~ — ©, while as x ~0, 1/x ow, and that at x = 0, 1/z Is 
undefined. The second class of cases in which the function is 
undefined occurs when the function is a fraction whose numerator 
and denominator both vanish for x =a; thus x/x* is undefined 
at x = 0, and so is x7/x ; another instance is sin x/x, when x = 0. 

In the case when the function is undefined at = a we examine 
the left and right-hand limits of f(z) as 2 + a, and determine by the 
method of the previous articles the two magnitudes 


Llim f(z), -~«—a, and Rlmf(z), xa 
If both these limits are finite and are equal, we write their common 
value as lim f(x), «—->a 


But if one or both limits are infinite, or if they are unequal, then 
we can assign no value to lim f(x”), « >a. 


24. Continuity and discontinuity of f(x) at x = a. 

Continuity or discontinuity at x = a can only occur when x = a 
is in a range for which the function is completely defined. The 
cond_t:ons for continuity are that 

Llm f(z, f@, Rlmf(z) («*#-a) 
are all three finite and equal. 

Discontinuity occurs either when one or more of the three are 
infinite, or when equality does not exist between the three values. 


25. Summary with illustrations. 

A brief restatement of the method in wh'ch number and function 
have been regarded in this chapter will now be set forth. 

Every number (a), whether rational or irrational, may be 
approached by sequences. Thus, in order of magnitude, we have a 
progression of numbers 

Lp ekis ss Piles eece(het aee Gh, .s- Qn, 11+ gy) Ga, 0 pene 


If f(x) is defined for every value of these two sequences, we have 
paired with the numbers in (A) a second set of numbers 


WG, uy Ge eee aie f(a),... J(@,)5 22 fla) fap: 
Now the two sequences in (A) define the same number, whether 
this number is rational or irrational. But the sequences 
fay) ? f(az) ’ f(a3) oy fan) se 
f(@), f(@e); f (as), .-s (Gy) eee 
even when they satisfy the test of Art. 17, do not necessarily define 
this part of the subject—perhaps the most important for his progress in the infini- 


tesimal calculus—to examine critically the values of cot x (~—>Q), secx (x > 90°), 
cosec « («->0); the case of tan x is discussed on p. 30. 


CHAP. II SUMMARY — yagi 


the same number. Supposing that the test is satisfied, the sequences 
define L lim f(x), x ~— a, and Rlim f(z), x ~a; further, we make 
the supposition that the same limits are found whatever sequences - 
are taken in (A) to approach a. 

In the case taken above f(x) is defined for x = a, and we have 
three numbers to consider 


Liim/f(z), f(a), Rlimf(z), x>a 


whether a is rational or irrational. But if f(x) is not defined for 
x =a, but defined for values in its neighbourhood, then we have 
only two quantities to consider, L lim f(z), R lim f(x), x >a. 


The following cases arise : 
I. The function may be continuous at x = a, in wh'ch case we 


have Llim f(x) = f@ =Rlimf@), r-a 
all three values being finite. 
II. The function may be undefined at x = a, and such that 
Llim f(z) = Rlm f(x), x>a 


both being finite. In th’s case the functions may be made con- 
tinuous by adding to the values of f(x) the value lim f(x), x >a, to 
correspond to x = a. 


III. The function may be discontinuous at x = a, when either 
one or more of the three quantities 


Lim/f(z), f@, Rlmf(x, xa 
is infinite, or two at least are unequal. 
IV. The function may be undefined at x = a, and we may have 
either Llim f(z)  Blim f(x) (« >a) 
or one or both of these limits infinite. 


Examples to illustrate II, III and IV are given below ; the case 
of I is so common that it needs no special attention. 


Examples to illustrate Case II. 
rg ot fiz) = a7/x («-—>0) 


It is impossible to evaluate f(x), when x = 0. But a*/x = x* for 
all values of x except zero. Now, when x—0, 2?/x—0, and when 
z— 0, «*/a—0; therefore 


Llim 2?/x = Rlima?/¢ =0 (x0) 
although the function is undefined at x = 0. 


* The student should notice that it is only legitimate to deduce a = c/b from ab = c, 
when b + 0, so that x = 2?/x follows from «? = x. a, only when x # 0. 


28 INFINITESIMAL CALCULUS CHAP. II 


Bix 2: f(a) = (w@ + v/a — 2)/(@ — 1) («> 1) 
The function is undefined atx = 1. We have 


x Ae 1-01 1-001 1-0001 


f(z) | 1-48 | 1-498 | 1-4998 | 1-49998 


and again 


x 0-9 | 0-99 0-999 | 0-999 


f(a) | 1:51 | 1501 | 1-5001 | 1-50001 


From this we might infer that 
Lihlm f(z) = Rlim f(%) = Lb +e 1) 
This result is obtained by writing 
je) — V2 = hoy + 2) 
(4/a@ — 1)(\/a + 1) 


which allows us to deduce the value of f(x) for all values except x = 1, 
by substituting for x in a/u +2 


“f/x + 1 
Ex. 3. f(x) = sina/x, x—+>0 (x being measured in degrees). 
Since sin (— x)/(— x) = sin x/a, it follows that 
Lhlm/f(#z) = Rlimf(x) («> 0) 
Also, from the table, 


x | 1 0:8 0-6 | 0-4 | 0-2 0-1 


sin x | 0-0175| 0-0140 0-0105 0-0070. 0:0035) 0-0017 


| 


we infer that lim sin w/z = 0-017 («+ 0) 


This result is confirmed by proofs given in Trigonometry, which show 
that lim sin6/0 = 1 (8-0) 


@ being measured in radians. 


The ‘answer, when @# is measured in 
degrees, is that 


lim sin a/x = 7/180 (x + 0) 


Examples to illustrate Case ITI. 


Ex. 1. The function {x} which is equal to the greatest integer which 
is not greater than x. 


Wemaverar 2@) cornea: {-d} =- 1, {st} = 0 


Now when « is in the range (0, 1], {a} = 0, when zis in (1, 2], {x} = 1, 
and generally when @ is in (n,n + 1], {a} =n. 


CHAP. II ILLUSTRATIONS 29 


The graph of the function, see Fig. 8, consists of a succession of steps 
of unit length, but in each of these steps the right hand terminal point 


Fie. 8. 


is omitted. This function provides examples of discontinuity at each 
integral value of a, for 


Llim {#) =n —-1 Wilimia)— nn @4 1) andint — 1 
Ex. 2. The function M(x), the mantissa of log, )%, and C(x), the 
characteristic of log; 9%. 


M(a) is discontinuous when w = 10”, for 

jobbet Awe | Ralint M(a) = 0 (i> 10") and M(10")=— 0 
Again, 

Mero Cita) == 9p 1 Kim Cie) = 2 (2% — LO") and C(L0%) 7 


Fic. 9a. Fic. 90. 


The graph (Fig. 9a) of M(#) consists of a series of lines starting from 
Ox and just reaching y = 1; as the origin is approached, these lines 
crowd closer and closer together and become steeper and steeper. 


30 INFINITESIMAL CALCULUS CHAP. II 


The graph (Fig. 9b) of C(#) consists of a series of steps, all on the 
right of Oy; as the axis of y is approached, these steps become shorter 
and shorter and further and further below Oz. In the drawing a 
portion of the axis of w is omitted in order to bring within the diagram 
an important feature of the graph. 


Examples to illustrate Case IV. 
rxpalt f(a) = 1fz 
Here we have 
Llim1/e = - Rliml/e = «0 («—-0) 
while f(x) is undefined at x = 0. 


x sa2s f(z) = 1f/a* (e@—= 0) 
We have Llim1/z2 = Rlim1/z? = 0 («—>0) 
The function is undefined at « = 0. 


Ex.00: f(x) = tana (x —-> in) 
Now Llmtanz = Rlimtanz = — 0 (#%-—> }nr) 
The value of tan x is undefined at ~ = $n. 


The erroneous (but convenient) statement that tan $x = © must ~ 
be regarded as an abbreviation of the statement that 


Llimtanz=o (x%-—- $n) 


26. Limit of the sum, product and quotient of two functions, when 
> 


The discussion separates itself into two parts according as the 
functions are defined or not for the value a = a. 


First, we take two functions, f(x) and g(x), defined at x = a and 
continuous at this value, that is, 


lim f(x) + f(@ lim g(x) >g(a) (wa) 
In this case we have, as x > a, 
lim [ f(x) + g(«)] = f@ + g(@ = limf(x) + limg(a) 
lim[f(z).g@] =f@.g@ = limf(a) .limg(a) 
lim [f(@)/g(@)] =f@/g@, = lm/f(z)/img@ 
provided that in the last case g(a) + 0. 


We have also to consider the case in which the functions satisfy 
the following conditions, when 2 > a, 


Llim f(z) = Rlim f(z) Llim g(x) = Rlim g(a) 


CHAP. II CONTINUITY By 


although f(x), g(x) (one or both) may not be defined atx =a, The 
following theorems hold 

lim[ f(z) + g(x)] = lim f(x) + limg(q) 

lim [ f(x) . g(x) ] lim f(x) . lim g(x) 

lim [f(x)/g(#)] = lim f(a) /lim g(@) 
provided in the last equality lim g(x) + 0. 

The proofs of these formulae present considerable difficulty ; the 
student will do well to defer them for the present. The theorems 
will be frequently quoted in the following chapters. Some considera- 
tions which may assist the student in apprehending the theorems and 
their proof are given in Appendix I at the end of the book. 


I 


27. Continuous functions. 

A function is continuous in a range of x for which it is defined, if 
it is continuous for every value of xin the range. Thus the poly- 
nomial function is continuous for any finite range; the reciprocal 
function (1/x) is continuous for any finite range included in[-o, 0] 
or in [0,0]; the tangent function is continuous for any range lying 
within any one of the ranges ...[ —47, in], [4x, 3x], [3x, $2] ...; 
while {x} is continuous for a range included in any one of the 
panes, Li(1;-2], 2,3 ]ix.2. 

‘It can be proved that the sum and the product of a finite number 
of functions which are continuous within a certain range are also 
continuous within this range. These theorems follow from the 
results quoted in Art. 26 with regard to the sum and the product 
of the limits of two continuous functions. It follows also that 
the quotient of two continuous functions is continuous, provided 
that the function which is the denominator does not vanish in 
the range considered. | 


Side by side with a function f(x) which is undefined for some value 

of x, say x = a, and is yet such that 
Lbim fay KRlim fiz) =) (¢—a) 

being also continuous at all other points of its range of definition, 
we may construct a function which is identical with f(x) at all 
points except x = a, and which at x = ahas the value /; this function 
may be termed an augmented f(x) ; and we can then assert the 
continuity of the sum, product and quotient of two such augmented 
functions on the same conditions as those laid down for functions 
defined and continuous at every point of their range. 


28. Examples to illustrate limits. 
Ex. 1. To find the limiting value, when x — 0, of 
J (1 + @ + uv) — 4/1 + @ — 2?) 
BS ae ig a ae 


32 INFINITESIMAL CALCULUS CHAP. I 


a/(1 + @ + x?) — o/(1 + @ — 2?) 
ae 
(l +a + 2%) —- (14+ @ — 2?) 
a[4/(1 + w + v7) + 4/(1 + @ — 2*)] 
2 
A/(1 + @ + x) 4+ 4/(1 + & — 2?) 


Now 


provided a + 0. 
The denominator — 2, when « — 0; therefore the given fraction — 1 
as its limit. 
As a verification we have, when z = 0:1, 
Wie set a eee], a/(1 + & + x*) = 1-:0536 
Lope — ae = 91-09 a/(1 + x — a) = 1-0440 
the value of the fraction for this value of x 
=)-0096 = 0-0 le 0-06 


Ex. 2. To prove that when n +o, > (2r + 1)/n*? —~1. 
“3 r=0 


Now 3) (2r + 1) = (n + 1), by the theory of arithmetical progressions. 


r=0 n 


Therefore > (2r + 1)/n? = 1 4+ 2/n + In? 


r=0 
and asn—~o,1/n—Oand 1 + 2/n + 1/n?—1. 


Ex. 3. A line BC of constant length slides between two fixed lines 
AB, AC of unlimited length, and B’C’ is a second position. To find the 
limiting position of P the wtersection of BC and B’C’, when the angle 
BPB’ decreases indefinitely. 


1 ceds 1). 


Let AD be perpendicular to BO, and let the angle BPB’ = 60. Then 
projecting CC’B’B upon CB, i 


CB = CC’ cos (r — C) + C’B’ cos 60 + B’BcosB 
Also BB’/ sin 69 = B’P/sin B, CC’/sin 66=PO’/sin C 
_ whence 2CB sin? $60/sin 686 = PC’ cot C — B’P cot B 
Now when 60 + 0, we obtain from this 
0= PCcotC- BPcotB 


CHAP. II EXERCISES 33 
BP tanB AD/BD_ DC 


Hence 

PC ~ tan @ Ge AD/DC Wissie 
Also BP+PC = DC+BD 
It follows that BP: = DC PG;=.BD 


Alternative proof. 

If BI, CI are drawn respectively perpendicular to AB and AC and 
meet in I, the point J is called the instantaneous centre of, motion ; 
its properties are discussed in a later chapter. From the discussion 
given there, IP is perpendicular to BC, and it follows that 


Dim=wei COLL bl i= PT tan Bb CP = PI tanC 
which leads to the same result. 


EXERCISES II 
1. Draw the graphs of the following functions 


] i i es x 
i il. ill. 
20 — 3 a+4 zx—l 
lV. 2a + 1 Vv. Ls mbes Vil. ae 
x —2 x(x — 1) 24) 
4 1 A 1 ; 1 
sini, cee SE PV ede Bleed cls it eee 
207 + 5x + 2 (2% + 5)? we+ue+il 


comparing the graphs with the graphs of the functions which are the 
reciprocals of the given functions. 


2. Determine the values of x for which each of the following functions is 
undefined, and discuss the functions in the neighbourhood of these values 


i. 1/(x + 1) ii. 1/4/(z% + 1) iii. (v? + w@ — 1)/(@ + 1) 
2 2 
_ eae oi ies ae Jaa Se ee ee 
3a + 3 a/(1 — x?) (x2 + a)? 
5 
Piipenne ee IN yii bOI Tie Codec! a 
a/(a — 1) xv + 384 — 4 
mee secer xi. log tan x xii. log tan?a 


3. State the range, or ranges, of ens of x for which the following 
functions are defined 


i. 4/(x + 1) ii, 4/(a? — 1) 

iii, 4/(4 — a) iv. «/{(a — 1)(a — 2)} 

v. [a(a — 1)/ax]? vi. w(x — 1)/[(x% — 1)*ex] 
vii. +/ {(a* — 1)/x} vill. / sin 

ix. 4/(1 — 2 sin 2) x. log x 

xi. 4/ log xii. log tan x 
xiii. log tan? xiv. log x/log(a? - 2% + 1) 
xv. log (sin % — cos z) xvi. log log x. 


oC. Cc 


34 INFINITESIMAL ‘CALCULUS CHAP. II 


4, Examine the limiting values of the following functions of « when 
a—~0; the answers may be obtained by writing for 2 a sequence of 
small values of x 


BEV. CLs) 20) 4/0 Lacey ee + 2Qa% — /(1 + 40%) 
x 2 af 
ey Lceiree) an (Leen) ea w+ 2-AW(et)) 1 


2 


ill. 


lv. 
x 3 x 4 
{Fe eds a\1 
Vv. Leap leawl mi Masri eis Tm ahi 
x 
abt. Hi rm S x 
pt ipreenlite Cig Mirus: eee Gene 
x 
eine fo : 1 — cosz 1 
Vill. 4 1a Rea REAR H 2L  S 
x x 2 
1 — cos 3x Tana. — aa 1] 
>. Ne Bren en) xls Satya ae — 3 
xii. sc cs mers xlii, cosec x — cotx—> 0 
x 3 


5. Prove that when n — o by a sequence of integers, 


i (1 +24+34+.. + n)/n?—~ 3 

ii. f14+24+3+.. + (n — 1)}/n?—4 
ili, (17 + 224+ 3? 4+... + n/n? ~3 

iv. {12 + 27 + 3? +... + (n — 1)}/n? — 2 
v. (13 + 23 + 33 +... + n3)/nt—+1 

vi. (14 + 24 + 34 +... + nt)/ni +1 


6. Represent graphically the sequence of points given by 


(1, a,) (2, Gy) (3, Gq) soe (1%, By). ce 
where 
i. ad, = (- 1)"/n ii. dg = nf[(n +1) iii. ag =[n + (- 1)"]/n 
iv. a, = (1+ 1j/n)” ve a, = 1+ (— 1)" vi. a, = (— 12 
and determine in each case the limit, or limits, of the sequence whose 
nth term is a,. 


7. From a point P on a quadrant of a circle, centre O, a perpendicular 
PN is drawn to OA, one of the bounding radii, and the tangent at P 
meets OA produced in 7. Prove that, as P approaches indefinitely 
near to A, 


i. lim PN?/AN = 20A ii. lim AN/AT = 1 
iii. lim PN?/(arc AP — chord AP) = 240A? 
8. Show also with the construction of the preceding example that the 


limit of the ratio of the area bounded by PN, NA and the arc AP to 
the area of the triangle PNT is 2: 3. 


CHAP. II EXERCISES 35 


9. A rod slides with its ends, A and B, one on each of two fixed axes 
at right angles, and AI, BI are drawn perpendicular to the axes at 
A and B. Show that the locus of a point P of the rod touches the line 
through P perpendicular to PI. 

Prove also that the limiting position of the intersection of two con- 
secutive positions of the rod divides AB in the ratio of OB? : OA?. 


10. A line AB is drawn to form with two fixed lines a triangle of 
constant area. Show that the limiting position of the intersection of 
two consecutive positions of AB bisects AB. 


11. On the rectangular axes of coordinates two points A, B are 
taken, and A and B vary their positions slightly in such a way that 
the perimeter of OAB is constant, the new positions being A’ and B’. 


Prove that AA’:B’B = AB +0B:AB+0A 


Show also that if the limiting position of the intersection of AB with 
its consecutive position is (zx, y), 
x:y = AB -OB:AB-OA 


12. ABC is an isosceles triangle whose base is BC; points P, Q are 
taken on CA, CB respectively so that AP = 2 BQ, and PQ meets AB 
producedin R. Given that the limiting position of R when P approaches 
indefinitely near to A, is S, prove that 


SB:AC = AC:2BC -— AC ~~ 


CHA Tilekee lil 
DIFFERENTIAL COEFFICIENT 


29. Definition. 


Let f(x) be a continuous function defined within an open range 
[a, 6b], and let xand x + h be values of the variable within this range 
—of these two values x for the present is fixed—and let 


Now, for all values of A such that 2 + hf is within [a, b], the 
numerator and denominator of F(h) are continuous ; consequently 
F(h) is continuous for every value of h in [- (x — a), 6 — x],* 
except h =0; if the left and right-hand limits of F(h) as 
h->0 are equal, this common limit is called the differential co- 
efficient of f(x), and is denoted by f(z). 


With the notation of the last article, we write the definition more 
succinctly 


iby Hates S =) Baa ira g —IO) _ pa ho) 


30. Illustrative examples. 
Ex. 1. To find the differential coefficient of x. 


aset(e)) =e. 
ao 
Then F(h) = {A ie ae he +h) 
h h 
Now, F(h) is undefined at h = 0, but at other values is identical 
with (2x + h); whence 


f(x) = lim F(A) (h > 0) = 22 


Ex. 2. To find the differential coefficient of x°. 


In this case 
3 3 2 2 
F(h) (a + a a h(3a* + au + h*) 


and i’ (x) = lim F(h) (kh > 0) = 322 


* The end-values of the range of A are obtained by giving to x + h the end-values 
of the range [a, b]; thus, 7 + h =a givesh = - (x - a). 


CHAP. III DIFFERENTIAL COEFFICIENT 37 
Ex. 3. To find the differential coefficient of 1/x. 


ciike Geng | agg a pee cael Meg at col oes 
h ha(x + h) 
Again, F(h) is undefined at h = 0, but at other values agrees with 
1 
a(x + h) 
it follows that f(#) = — 1/x 


Ex. 4. To find the differential coefficient of \/x = x, 


SO Cae t Sees eee Ce ieee: 
TR ENS aaiuaat Neate NCCE SIENA 
h 
hh (@ + h) + v/a} 
whence F(a) = lim F(R) (h> 0) = 5 = 


Note that the range of this function is (0, © ]; its range for the purpose 
of differentiation must be open, and is taken as [0, ©]; hence the 
differential coefficient has no value at x = 0, which is confirmed by the 
form of f’(z). 


EXERCISES III (a) 


Differentiate the following functions, stating the range of values of x 
of the differential coefficient when the range is restricted 


Lee orice DEN ed | 3. 3227 + a — 4 
4, x 5. (2 + 1)-? 6. 2 

7. (Qe + 1)? Sa(at ea) a! 9. 4/(% + 1) 
10. 4/(3 — 2a) Ll. (a2 + 1)-? 124 (2) eal es 


31. Remarks upon the definition of a differential coefficient. 


The left-hand limit of F(A) is called the lefé derwative * of f(x) ; 
also R lim F(A) (kh ~ 0) is called the right derwative. A function 
may have a left and a right derivative and yet have no differential 
coefficient ; this case occurs when 

Llim F(h)  Rlim F(h) (2 > 0) 

An open range was selected for the definition of f(x) in order to 
secure that f(z) might have a differential coefficient at each point of 
the range of its definition. If the range of definition had been 
closed, f(x) could have had no differential coefficient at either of the 
closed ends ; thus, if the range of f(x) is (a, 6), f(x) might have a 


* The word derivative is not used in this book as synonymous with differential 
coefficient ; it is only used when qualified by the words ‘right’ or ‘ left.’ 


38 INFINITESIMAL CALCULUS. CHAP. III 


right derivative at 2 = a, and a left derivative at x = 6, but could 
have no differential coefficient at either end-value. 

Again, the cases in which Llim F(A), Rlim F(A) (hk +0) are 
infinite are excluded, because the equality of two infinities cannot 
be asserted. It follows that the differential coefficient is never 
infinite. 


32. Notation. 

The symbol (’) applied to the functional symbol, which has been 
adopted to indicate the differential coefficient of f(x) is convenient, 
but other symbols are also used, and the development of the subject 
is bound up with the adoption of a second symbol which has proved 
more important than the dash. 


If we write Te ye) 
and suppose that when the independent variable is + dx(= ax +h), 
the dependent variable is y + dy; then we have 
y + oy = f(x + dx) = f(x +h) 
fe eh) = fa) veg 
cae eran DTI Bar 
Here 6x is a small increment of x and dy is a small increment of y ; 
the limit of dy/dxz, as da +0, is the differential coefficient of f(x). 
This limit is written dy 
dx 
The student must note that this expression is no fraction ; it is 
a single entity. Comparing the two notations, we have 


and 


y = f(x) 
, dy» 
side by side with Fie F(x) 


33. Geometrical illustration. 
Let OM =2 MP =y = f(x) NQ = f(x + dz) = y + dy 


Then Fh) = a = a = tan RPO = tan 7sP 

Now, if we take upon the arc of the graph of the function a 
sequence of points Qj, Qs, Qz, ..., Which has P as its limit, we have 
a sequence of chords PQ,. PQs, PQs, ..., which, on being produced, 
meet the axis of x in a sequence of points Sj, S,, Sg, .... Now the 
secants Q,PS,, Q,PS., Q3PSs, ... form a sequence whose limit is the 
tangent at P; hence the limit of S8,, S,, S3,...is 7’. It follows that 

dy 


ail tan 27P 


CHAP. III GRADIENT OF A CURVE 39 


If the tangent at P is parallel to Ox, S,, S,, S,... tend to infinity ; 
in this case the student may replace the points of the sequence 
S, S,, Sg, ... by the points in which the secants meet Oy. 


34. Second notation. 


An extension of the notation for a differential coefficient may 
now be made. We have 


d , 
y=fc) = =f'@) 
These may be replaced by 
df(x) 


dz 


f (x) 


or, if we regard differentiation as an operation which, when applied 
to the function, produces its differential coefficient, we may detach 
the symbols and write 


d / 
ed = f'(x) 


This form of the notation is particularly useful when the function 
to be differentiated is long. 


35. Rules for differentiating the sum, product and quotient of two 
functions whose differential coefficients are known. 


The rules to be proved may be stated thus 


d } d d 
I. dy + g(x)] =f(z) +9 (a) = Fed) + A I) 


40 INFINITESIMAL CALCULUS CHAP. IIt 


Il. © (fe) (2) Wea iole) gaa) g (x) 
d d 
= Th g(x) + f(x) ia 9) 


te d 
rr, 2 £2). $e) ote) ~ Fle) Fe) _ al 8 TI ae 


"dx g(x) [g(a) ? [g(x) ? 
In the last formula it is prescribed that g(x) is not zero for 
the value of x considered. 


I. T'o prove that Oy (2) + g(x)] = f(x) + g/(2) 


Let s(x) = f(x) + g(x); s(x) being the sum of two continuous 
functions, is continuous for any range for which f(x), g(x) are 
continuous. 


sz +h) — su) = fla + h) — f(x) + ge +h) - g(a) 


sisal Maan Meals Came Car eae) 5, Gerardo) 300 
h 7 h h 
beep eee i NC raga IER a) — f(x) 
Ch) g(w + 2 — g(x) 
Then S(h) = Fh) + Gh) 


Here again F(h), G(A) being continuous except at h = 0, the same 
holds for S(h) ; also 


lim Sh) t= 7s Ge) lim Eh) tee 7G) lim G(h) = g(x) (h +0) 
and by the results stated in Art. 26, 
lim [F(h) + G(h)] = lim F(A) + im G(h) (h + 0) 
Thetis, lim S(A) = hm F(A) + lim G(A) 
or S(t) ea TA og (a) 
This is the required theorem. 


It is easy to prove the theorem for the sum of a finite number 
of functions, and also to show that } 


d ; / 
F [fe - 9@l = £'@) - 9) 
II. T'o prove that 
© a) - 9fa)\ = f'(@) gle) + fe) g@ 


oHAP. tit DIFFERENTIATION OF A PRODUCT 4.1 


Let p(x) = f(x) g(x), then p(x) is continuous for any range for 
which f(x), g(z) are continuous ; hence 


pix +h) - p(x) =fa+hgawt+h) — f(x) g@) 
=[f~@ +h) - fiw] ga +h) + f@wl[og@w +h) - g(2)] 
It follows that 


gle +h) + fea) 2 +8) — 9 
= Fh) gw + h) + f(x) Gh) 


Now P(h) being made up of the sum of products of functions of h 
which are continuous over ranges which do not include h = 0, P(A) 
is continuous except at h = 0. By the results of Art. 26, 


lim [F(A) g(a + h) + f(z) G)] = lim [F(h) g(a + h)] + lim[ f(x) Gh] 
lim P(A) = lim F(A) . lim g(a + h) + fiw) lim Gh) (kh + 0) 
= lim F(h) . g(x) + f(x) lim G(A) 
and, as lim P(h) = p(x) (kh +0), we have the second theorem, 
p(x) = f(x) g(x) + f(x) g(x) 


We can deduce the differential coefficient of the product of any 
finite number of functions of x. Taking the case of three functions, 
and making a slight change in the notation in order to familiarise 
the student with the new symbols used, 


aloes ‘ — f(x) 
Ly h 


uid) = a yay) = yay + 
ni YY 243 Yr - YxY3) = -YoY3 + 17, le Y2Y3 


dy, on ay 
= YoY; — ae z° vs(y2 ike + Ys abe 

dy, dy» 2 Ys 
= Ys a + Y3Y1— An YiY27,. aie 


Another useful deductions is obtained by writing g(x) =C€ (a 
constant), in which case g’(x) = 0, and 


d 
5 LCs] = Cf @) 
III. 7'o prove that 


d fax) _f@g@ - fog 
dx g(x) [g(x) |? 
Let q(x) = f(x)/g(x), then g(x) is continuous over any range for 


42 INFINITESIMAL CALCULUS CHAP. Ill 


which f(x) and g(x) are continuous, and in which g(x) does not 
vanish. We have 
Te teri) eat (ey 
q(z + h) — g(x) = FERNY An EES 
_ f(a +h) g@) — fix) g@ + h) 
i; g(x) g(a + h) 
_ [f@ +h) - f@)lg@) - f@lg(e + — g()) 
. g(x) g(a + h) 
qe +h) — g(x) _ FM) g(a) - f@ GQ) 
h g(x) g(a + h) 
Now g(x) + 0 and A must be taken so small that g(a + h) + 0 for 
any small value of A; with these prescriptions Q(f) is continuous 
except ath = 0. Also whenh > 0, 


fin Fl). 9@)_ — fa) GM) _ lim [F@) g(e) ~ fle) GM) 


Again, Q(h) = 


g(x) g(x + h) g(x) lim g(x + h) 
_ lim F(A) . g(x) — f(a) lim Gh) 
eles g(a) g(a) 
1, _ bf (t) g(x) — f(x) g’(2) 
whence q (x) ai (g(x)? 


The above results are of the greatest importance and must be 
committed to memory. For those who may prefer words, the 
following statements are given : 


I. The differential coefficient of the sum of two functions 18 equal to 
the sum of their differential coefficients. 

II. The differential coefficient of a product of two functions ts the 
sum of the differential coefficient of the first function multiplied by the 
second function and the first function multiplied by the differential 
coefficient of the second function. 

Ill. The differential coefficient of a quotient of two functions is a 
fraction whose numerator is the difference between the product of the 
denominator by the differential coefficient of the numerator and the 
product of the numerator by the differential coefficient of the denomi- 
nator and whose denominator is the square of the denominator. 


36. Illustrative examples. 


d d d d 
i Seo AAS ap ee PO paeety  Ue Saas 


i xo RSQ sate ayl Wail BATS a Le 
dx x x a) 


cHaP.1t DIFFERENTIATION OF A POWER 43 


d 3 d d d d 
° . a. b 2 ad = — f 3 —— 2 aie ee 
Bex. oo oe (ax® + bx* + cx + a) a ax? + 7 ba? + ae cx + qa 


d d d 
Me rie Serio? ey 
a nie O reas Ga 


3ax* + 2ba +c 


I 


Ex, 4, 


Ee tees + b)(cw + d)] = (cw + d) ue (ax + b) + (aw + ae ea + d) 
daz da dx 


I 


(cx + d)a + (ax + b)c 
2acxe + ad + bc 


d d 
ca eee. Cth dae ii Gi aa ae 
dxcx +d (cw + d)* 
a(cw + d) — c(ax + b) 
i. (cw + d)* 
_ ad — be 
~ (cx + ad)? 


37. Differential coefficient of x", when n is integral. 


First, we shall prove by the process of mathematical induction 
that, when v is a positive integer, 


i TDI 1 (tooth 
dx 
The theorem is true for n = 1, 2,3; as we have already shown 
in Art. 30, from first principles, that 
d Ae 0 d Pl ee 1 d 4 aa 2 
melee an = 2% ae = 8 
Let us assume that the theorem holds for the index n — 1, that 
is einat d 


ae yn) pial n—-2 
RP (n — l)ax 
Then, by the result II of Art. 35, 
d ee oe d n-1 —_ —1 d d n—-1 
ee ee eoargee Ape Vek pe 


SECRETE READ ry Lge ip Gal aay (ee Gre 


The theorem assumed for the (n — 1)th power of x has been 
proved to be true for the nth, and it holds for n = 1, 2,3; it therefore 
holds for all positive integral indices. 


44 INFINITESIMAL CALCULUS CHAP. m1 


Secondly, let n be a negative integer, say m = — m, where m is 
positive ; then 


Cot = © lam 21-1. Fam | atm 


sy — 
dx da x™ dx d. 
- mam-l 
= San = — mae-"1 = nz! 


The theorem holds for all integral values of n. 


38. The differential coefficients of sin x, cos x, tan x. 

The unit of angular measurement is the radian. 

ety) cin wz, 

fle +h) - fixe) _sin@ +h) — sine — 2cos(x% + $h)sin$h 
h MS h a h 
F(h) = cos(x + th) sin $h/th 
Now lim F(A) = limcos(a# + 4h) lim (sin $h/th) (h +0) 
= COS 1 


wheretore f' (2) = Cos “01 © sin x =cosxz 


Bean, det f(a) cose -ythen 


fist lh) = fG) |) cos: (x bik) = "C08 2a Wea Z sini th) sin th 
h ze h Fite h 
F(h) = — sin (« + 4h) sin $h/th 
and it follows that & cosx =-— sing 


Thirdly, let "= tan 2 — sin 2/cos wathen 


ad. : d 
cos 7=— sin # — sin %-- cosx 


dy da da 
da cos?2 
cos2% + sin2x 4 
= ee ee 
cos2x 


39. The differential coefficients of cot x, sec x, cosec x. 


d cos2x — sin v7 sin ® — cos # cos x 
—cotzr = —— = as = — cosec*x 
da dx sin x sin?x 
cos ay Lae 
C1 CO : 
d he ik da dx Sin x 
—sec x = — he Sp A 5, = tan xsec x 
da dx cos x cos?x cosa 
d ee aH 
— cosecx = — -= = — COL  Cosec X 
dx dx sin x 


CHAP. III ILLUSTRATIVE EXAMPLES 45 


40. Examples on differentiation. 


xr obe ve Ysera oil 2 

dy ; 

=~ =siInwv + xcosx 

dx 

re 

[Dee WPS PS 

y tan x 

dy tanz —axsec*x sinwcosx — x 

dx tan2x aa sin? x 

x + sin x 
Ex.3, 9 y = ——— 
“x — sine 
dy (1+ cos)(x — sin x) — (1 — cos2)(x + sin 2) 
dx (w — sina)" 
_ 2(% cos % — sin x) 
(x — sin x)? 
Bx 4. = sins 
EPeINere sil e-? Sill a (7 1aCLOlLs) 

dy ae 

dy 7 COS sin™-ly + cosasin"-lx + ... (n terms) 

fr 


= ncos x sin” ly 


Ex. 5. Two lines OA, OB at right angles are measured, and their lengths 
are given as 36, 48 yards respectively. If a possible error of 1 foot has 
to be allowed for in the measurement of OA, show that an error of 7-2 inches 
may occur in the length of AB deduced from the measurements. 


Now, let OA = a, OB = band AB = ec; then 
ce = a? + 6 

If we allow an error 6a in the measurement of a and a consequent 

error 6c in c, we have 
(c + dc)? = (a + da)? + 6 

On subtraction, 2cdéc + (dc)? = 2ada + (da)? 
de 
ba 


We want the ratio of dc/da when 6a and consequently écis small. This 
ratio is given by omitting dc compared to 2c and 6a compared to 2a, and 
we obtain Se | 


° 3a 


or (2c + dc) — = 2a + da 


a 
whence éc = —da 
Cc 


Substituting a = 36, c = 60, da = 12 inches, we get the required 
result, 


46 INFINITESIMAL CALCULUS CHAP. III 


Ex. 6. Criticise the following passage: Eridu, a port of early Babylonia, 
lies now 125 miles from the sea. If the present rate of advance (of the sea 
coast), about a mile in thirty years, may be taken as an average, Hridu 
may have been mud-bound about 1800 B.c. 

The writer assumes that by the deposit of the river Euphrates the 
coast has advanced at a uniform rate during 3700 years. The develop- 
ment of a delta is perhaps better represented by the growth of a sector 
of a circle; we shall calculate the rate at which the boundary of such a 
sector advances when the area increases at a constant rate, the angle 
of the sector remaining the same. 

Let A be the area of the sector at time #, r its radius, « its angle; then 

A = ar? 
If 5A is the increase of A, and 6r of r in time &, 
A + 0A = $a(r + dr)? 
whence 6A = ardr + $a(dr)? 
= ar dr, nearly 


Now 6A/éé is constant, if we assume that the amount of silt is the 
same each year and is spread evenly over an area of equal depth. Hence 
dr/dt, the rate of advance of the sea coast, varies inversely as r, which 
means that the conclusion in the passage needs further justification. 


EXERCISES III (s) 


1. Differentiate the following functions 


Fk | es .. 2 — a2 ve oe 
i. Uy is oie ee Jitsu eee 
a+z2 a? + a2 ys» 
. 2 + a? l+at a . iy Soar pies 
1 Vine ieee ches Vv. Viv oe eee 
x — a? l—a2+ 2? l+a2+ta 
a 2 
Dy ee seth muP ae EE 
lx l+2 1 — 29 4a 
fs 2 
a l1-—-2&+z2x a (~ + 1)(w + 2) xii. x+3 
oe x+3 — (@ + 1)(x% + 2) 
SSE Tras wacom orem Ce Tun ONC ed xv, M23 eS 
(« — a)(x — b) (~ — 3)(w% 4+ 2) (x + 1)(@ — 2) 
. gen — qn és fa LP 
2,01) Ageia SS ae AEE Xvi. (lL e/a) 4/2) XVI. eee 
5 (+ yay = v2) xviii, LEV? 
, 1 — ; : 
xix. 1- vx Xx. sin?x% cos x XXi, 2 COS 2 
1 + +/x 
XXil. u XX. a Sin xxiv. tanx + secx 


sin x 1 — sinz 
sin 2 — COs & .. ginx + coszx 
XVI ee eee 


go Va SCC. ts COSEG arte Vinee ee - 
sin x + cos x sin vw — cosx 


CHAP. III EXERCISES 47 


2. Prove that the error made in estimating the length of the circum- 
ference of a circle, radius 7, due to an error of 67 in measuring the radius 
is 2m or. 


3. A circular track measures 398 yards close round the inside. 
Show that its measure 1 foot out from the inside is approximately 
400 yards. 


4, A square plate is contracting while cooling; and when its side 
is 10 inches, this side is diminishing at the rate of 0-1 inch per minute. 
Show that the area is decreasing at the rate of 2 sq. inches per min. 


5. The radius of a circular plate is 3 inches, and is increasing at the 
rate of 0-01 inch per min. Show that the area is increasing at the rate 
of 0-19... sq. in. per min. 


6. One end of a ladder, 20 feet long, rests on the ground at a distance 
of 12 ft. from a wall against which the other end rests. The ladder is 
raised, the lower end being shifted 4 in. nearer the wall. Prove that 
the upper end is raised about 3 in. 


7. Show that the errors made in calculating (i) the surface, and (ii) the 
volume of a sphere, radius 7, due to an error of 6r in the measurement of 7, 
are 8nré6r and 4nr26r respectively. 

If the radius is 10 in. and the error in measuring it is 0-1 in., show 
that the percentage errors in the surface and volume measurements are 
2 p.c. and 3 p.c. respectively. 


8. The volume of a cube is increasing at a constant rate. Prove that 
the surface-increase varies inversely as the length of the edge. 


9. A rectangular block of ice rests on a non-conducting base and 
melts on the surface exposed to the air so that a layer of thickness 
0-1 in. is converted into water each min. Show that when the height 
is 5 in. and the breadth and width both 3 inches, its volume is decreasing 
at the rate of 6:9 c, in. per min. 


CEA Peles Rai 
THE SIGN OF THE DIFFERENTIAL COEFFICIENT 


41. Derived function. 

A function f(x) which has a differential coefficient for every value 
of x in an open range is said to be differentiable in that range. The 
totality of the differential coefficients of f(x) constitutes another 
function f(x), called the derived function of f(x). 

In this chapter we propose to consider some problems relating to 
f(x) which can be solved by a study of the properties of f’(~). These 
problems will be discussed with, and without, the help of graphs. 
Graphical methods will probably appeal more directly to the beginner 
than those which are based on purely arithmetical considerations, 
but if the student wishes to obtain a firm grip of the subject, he 
must master the more general proofs. Possibly the beginner may be 
wise in deferring these to a second reading of the subject. 


42. Meaning of the sign of f’(x). 

We shall show that, if f(a) ws positive, f(x) is increasing as x 
(increasing) passes through x =a; while if f’(a) is negative, f(x) is 
decreasing, as x (increasing) passes through this value. 

We recall that in defining a differential coefficient we introduced 


a function Bs 
F(h) ee fla 1 a f@ 


in which the constant a plays its part in the construction of F(A), 
though it is not explicitly mentioned in the symbol F(A). Further, 
though F(h) is undefined at A = 0, its right and left-hand limits are 
equal at this value. 

We now examine two ranges of the variable h, one on the right of 
x = a, in which case his positive, and the second on the left of x = a. 
Now, since Rlim F(h) = f(a), h ~0, the terms of the sequence 
which gives R lim F(A) must, from and after some value of h, have 
the same sign as f(a). If then we discard terms which occur 
before this, we are left with a sequence of values of h, namely 
hi, ho, hs, ... , to which corresponds a sequence 


F(A,), F (hs), FAs), ewe 


CHAP. IV STATIONARY VALUES 49 


all the terms of which have the same sign as its limit f’(a). Hence, 
if f’(a) is positive, F(A), F(h,), F(As), ... are all positive. It follows 
at once from the definition of F(h) that 


fa+h) >f@ fa +h) > f@, ... 


That is, when ~ is just greater than a, we have 
f(x) > f@ 
Again, let us approach the left-hand limit of F(h) by a sequence 
~k,, — ka, — hy, ... 
such that every term of the sequence 
F(- k), F(- kh), F(- &), ... 
has the same sign as f(a) ; or, taking — k as a typical term, 


fa —k) - f@ 


has the same sign as f’(a). Thus, if f’(a) is positive, fia — k) — f(a) 
is negative, therefore f(a — k) < f(a); that is, when 2 is just less 
than a, f(x) < f(a). 


Joining together the two statements as to what happens just 
before and just after x = a, we see that, as 2 passes through the 
value x = a, f(x) is increasing, if f’(a) is positive. 


By the appropriate changes in the above statement we can 
show that, if f’(a) is negative, f(~) decreases as x passes through 
ee 


The converse of the theorem is also true, namely, that, af f(x) is 
increasing as x passes through « = a, f(a) is positive; while if tt is 
decreasing, f(a) is negative. The proof which is left to the student 
consists in a rearrangement of the facts stated above. 


43. Stationary values of f(x). 

If f(a) = 0, we cannot say that the function is increasing or 
decreasing ata =a. Whenf’(a) = 0, wesay that f(a) isa stationary 
value, and we leave to a later section the important discussion of 
the various ways in which f(z) may behave near such a value. 

To find the stationary values of f(v) we have to solve the equation 
f(x) = 0; the roots of this equation determine the values of 2 which 
give the stationary values of f(z). 


44. Graphical interpretation of the sign of f’(x). 

The function whose graph is given in Fig. 12 is now considered. 
We suppose that it is described in the direction ABC... , in which 
the independent variable is increasing. From A to B the curve 
is on the up-gradient, and from B to C on the down-gradient, and 

C.C, D 


50 INFINITESIMAL CALCULUS CHAP. IV 


soon. Taking P a point on the arc AB, and drawing the tangent 
at P, we have 


dy ; 
tan 7) Pe os (att) 47 {2) 


Since x7’P is acute, its tangent is positive; therefore f’(x) is 
positive. It follows that at all points on the up-gradient between 
A and B, provided neither A nor B is included, the differential 
coefficient is positive. | 


Fic. 12. 


Again, at Q, a point on the downhill section BC, «TQ is obtuse 
and tan aT’Q = f’(x) 


thus the differential coefficient is negative at every point on a 
downhill section. 


The geometrical illustration presents almost intuitively the two 
theorems (i) that, when f’(a) is positive, f(x) is increasing as x passes 
through x = a, and (ii) that, when f(x) is increasing at 2 = a, f’(a) 
is positive. 

Now at A, B, C,..., f(x) is zero, but the curve given does not 
afford a complete analysis of the problem of the behaviour of a 
curve at a stationary point. 


45. Maximum and minimum values. 


We shall discuss only maximum and minimum values which 
are also stationary values. 

As a definition we take : af, as x passes through x = a, which gives a 
stationary value of f(x), f(x) changes from increasing to decreasing, 
then f(a) 1s a maximum stationary value of f(x) ; while, of f(x) changes 
from decreasing to increasing, then f(a) 1s a minimum stationary 
value of f(x). 


In the neighbourhood of a stationary point f’(x) behaves generally 
in one of the following ways indicated below, the exception being 
when f’(a) remains zero on one or both sides of # = a (a case which 
we need not particularise). 


CHAP. IV MAXIMA AND MINIMA 51 


Dim? Cae On toe Ch 
ea + -+ Cate 
(Ore a 
4 ae ae ao 
Hoya aE ere ign ie 


In case I, f(x) is increasing as x increases to 2 = a, and goes on 
increasing after x = a, but its rate of increase is zero at w = a. 

In case IV, a similar explanation can be given. 

In case II, f(x) increases as x increases to x = a, and decreases 
after « =a; this is the case of a maximum stationary value at 
eee, 

In case III, f(x) decreases as w increases to x = a, and increases 
after x = a; this is the case of a minimum stationary value. 


The geometrical interpretations of all four cases will be given 
below; but we can deduce that the condition for a maximum 
stationary value of f(z) at x =a is that, as x passes through the 
value x =a, f'(x) changes from positive to negative; while for a 
minimum stationary value, f’(x) changes from negative to positive. 


Again, if f’(~) is continuous, there are no maximum and minimum 
values which are not stationary. For at (say) a maximum given 
by x = a, f(x) changes from increasing to decreasing, and therefore 
f(x) changes from positive to negative ; it follows, since f’(x) is 
cont nuous, that f’(a) = 0, that is, f(a) is a stationary value. 

There are maximum and minimum values of a function which 
are not stationary values. Thus the apex B of a triangle ABC is the 
point on the line ABC, which is at a maximum distance from AC, 
but the perpendicular PM from a point P on ABC is not a 
differentiable function of the variable AM which determines its 
position, and therefore PM has no stationary value. 


46. Relation between the graphs of f(x) and f’(x). 

This relation is so important, in view of the future use to be made 
of these graphs, that it is worth while side by side with the graph 
of f(x) to draw the whole of the graph of f’(x), although for the 
particular purpose of the problems discussed in this chapter, it is 
generally enough to draw only those parts of the graph of f’(a) 
which are near to Ox. 

In the diagram (Fig. 13) the graph of f(x) is PBCD... , while that 
of f’(x) is indicated by a dotted line. The points B, C illustrate 
maximum and minimum values; thus B indicates a maximum 


52 INFINITESIMAL CALCULUS CHAP. IV 


stationary value, C a minimum stationary, while D illustrates 
case I of the last article and F illustrates case IV; for f’(x) is positive 
near D’, while f’(~) is negative near F’, vanishing at both D’ and F”. 

Describing the graphs in the direction of 2 increasing, we say 
that, as P’ is above Oz, f(x) is increasing at P; as B’ is on the 
axis, f(x) has a stationary value at x = 6; again, from B to C 
f(x) is decreasing, the graph of f’(%) is below the axis ; from C to #, 
f(z) does not decrease, the graph of f’(~) not being below Ox; there 
is, however, a point of the graph of f’(x) on the axis which corre- 
sponds to the stationary value of the function at D. 


Fre. 13. 


It is clear that at a maximum stationary value the graph of 
f(x) crosses Ox from above to below ; this is called (on the analogy 
of astronomy) a descending node ; * while at a minimum stationary 
value the graph of f’(x) crosses Ox from below to above, that is, has 
an ascending node. At other stationary points the diagram suggests 
that the graph of f’(x) touches Ow. 


47. Examples of maxima and minima. 
Ex. 1. A quadratic function has values 3, 5, 4, when x = — 1, 0, 1 
respectively, to find its greatest value. 


Let the function be f(x) = aw? + bu +c 
The conditions give 
a-6b+c=83 C= 5 atb+c=4 


These are satisfied by a = — 14, b = }, c = 5; hence 
f(w) = — 307 + du + 5 
fe) = - 3a + 4 


The stationary value is given by x = }; in the neighbourhood of 
this value, if  < }, f(x) is positive, while if a > 2, f’(x) is negative. It 
follows that f(}) isa maximum. The required maximum = 5,1. 


* Node implies the crossing of two lines on the diagram, 


CHAP. IV EXAMPLES 53 


Ex. 2. It is required to divide a line AB at P so that AP? + p. BP? 
ts a minimum. 
Lev APi=]o b= a — x. 
f(x) = 2% + pla — x)? = 2? + p(a® — 2ax + 2?) 
f(x) = 2x + p(— 2a + 2x) 
" = 21 + p)x — 2ap 


Now the graph of f(a) has an ascending node at 
wz = ap/(1 + p) 
hence the required point of division is given by this value of a. 


Ex. 3. To find the stationary value of 


4 
YA 
- 1l+2 
and to show that it is a minimum. 
Let (cle eee : 
Aare La 
4 2e(1 + x)? + 4 
“e) = 9 ——_—_—— = 
UNG ieee tdiee  gyaK ae (1 + ap 
_ go (@ + 2)(e? + VD) 
(1) 0a)® 


The ranges in which f(x) and f’(x) are defined are[ —- ©, — 1][-—1,]. 
The stationary value occurs in the first of these ranges, and is given 
by x =— 2; near this value, when x< — 2, /’(x) is negative, and 
when «> — 2, f(x) is positive. Therefore the minimum stationary 
value is f( — 2) = 8. 

It is to be noted that f(a) has many values less than 8 in the range 
[-— 1, o]. 


Ex. 4. The power required to propel a steamer in still water varies as 
the square of the speed. Prove that the most economical rate at which tt 
can travel against a current is equal to the speed of the current. 


Let V be the rate of the current, v that of the steamer relative to 
the land; V + v is the rate relative to the water. The power varies 
as (V + v)*, and the consumption of coal varies jointly as the power 
and the time; the time is equal to //v, where / is the length of the 
journey. We must therefore make (V + v)?/v a minimum. 


Let f(v) = (V + v/v = v + 2V + V4/v 
fv) =1 - V2/e 


The only values of v which concern us are positive ones; v = V 
gives a stationary value of f(v), and this is a minimum, because 
for values of w near to V which are <V, f(v) is negative, and for 
those > V, f’(v) is positive. Hence the most economical rate of progress 
is when v = V. 


54 INFINITESIMAL CALCULUS —_ CHap. Iv 


Ex. 5. To find the dimensions of an open can which is to contain 1000 
c.cm. when the smallest amount of material is used. 


It is not inconvenient to solve first the general case. With the 

usual notation, we have 
V = 27rh S = mr + Qnrh = xr? + 2V/r = f(r), suppose 

Now f(r) = 2nr — 2V/r 
The stationary value is given by zr? = V, that is, r = (V/r)*. Near 
this value when ris < (V/x)®, f(r) is negative, and when r is > V(/n)?, 
f(r) is positive. Hence this value of r gives the minimum of 8. 

The dimensions of the can of smallest surface are 

r=(V/n)i h=V/[rr=(V/r)i S§ = 3(nV)3 

Writing V = 1000, b= 9045.57 cm, 


Ex. 6. To find the dimensions of the cylinder of greatest curved surface 
that can be inscribed in a sphere. 

The curved surface of the cylinder = 2zrh, and since it is inscribed 
in a sphere of radius a, 472 + h2 = 4a2 


Instead of finding the maximum value of 8, we determine the maximum 
of S? or the maximum of r?h? = f(r), suppose. Hence 
S(r) = r2h? = 4r2(a? — 1?) = 4a?r2-— 4r4 
S'(r) = 8r(a? — 2r?) 
Since r is essentially positive, we consider only the stationary value 
given by r = a/,/2. Now as r passes through this value, f’(r) changes 
from positive to negative. Hence, r = a/,/2 gives a maximum of f(r), 
and therefore of 8. 
The dimensions of the required cylinder are 
r= a/r/2 h = av/2 hat ite 
This example can be solved also by taking an auxiliary angle 0, such 
that r = asin 0, h = 2acos6; then S = 4za*sin 8 cos 0 = 2za? sin 20, 
and the maximum of § is obviously given by sin 20 = 1. 


Ex. 7. To find the dimensions of the right cone of greatest total surface 
that can be inscribed in a sphere of radius a. 


By revolution about Ow the diagram determines a sphere and the 
inscribed cone vertex A. If P is (x, y), we have 
(2 —a?? + 7 = a? 
The dimensions of the cone might be expressed in terms of 2 or y, 


but it is simpler to select as variable OP = z; this choice is made after 
trial, and secures the expression of the surface as a polynomial in z. 


Now PN =r, AN =h, AP =s8 in the cone-notation; we have 
r=y,h = 2a— 2, s* = 2a(2a — x), y* = 2(2a — =), 22 = 2az, and 
S = my” + mys 
S/x = x(2a — x) + (2a — x)a/(2az) 
(2a — 2#/2a)(z?/2a + z) 


I 


4 


CHAP. IV EXAMPLES 55 


Differentiating, 


£ (S/n) =—- z/a(z"/2a +2) + (z/a + 1)(2a - 22/2a) 


— (228 + 3az* — 4a%z — 4a3)/2a? 
= — (2 + 2a)(2z2 — az — 2a?)/2a? 
— (2 + 2a)(z — 2,)(z — 2,)/a 
where z, = }a(1 + 1/17), 2 = ja(l — +/17). 


I 


| P- 


Fic. 14. 


Now as z is essentially positive, we consider only the stationary 
value given by z = z,; the graph of the differential coefficient shows 
that z = z, is a descending node; therefore 2 = z, gives a maximum 
value of 8S. By substitution, we find 


r = jar/(190 + 144/17) = 0-98... a 
h = },a(23 - 4/17) = 1-18...4 
Ex. 8. To find the shortest line passing through a point in the first 


quadrant whose coordinates are (a, b,) and having its ends upon the positive 
parts of the axes of coordinates. 


Let h and k be the intercepts on the axes; we require a minimum 
value of 1/(h? + k?), where h and & are connected by the relation 


a/h + b/k = 1 
We shall seek the minimum value of 


h? + hk? = k* + a®k*/(k — b)* = f(k), suppose 


, 2a*kb 
Now I'(k) = 2k — (k — 6) 
9, (& — 5)8 — at 
= 2K a eh eee 


The only values of k which we consider lie in [b, © ]; therefore we need 
only look at the value 


k = b + (a2b)} = b3(a> + 03) 


56 INFINITESIMAL CALCULUS CHAP. IV 


It is easily seen that /’(k) has an ascending node at this point. There- 
fore f(k) is a minimum, when 


k = bias +b) = hh = ad(at + 58) 
The shortest line is = (a + b3)%, 


The above provides the answer to the problem of finding the longest 
thin beam which can be taken in a horizontal position round a rectangular 
bend in a passage, the widths of the two parts of the passage being 
aandb. This problem is solved kinematically by noticing that in the 
critical position the line joining J, the instantaneous centre of motion of 
the beam to (a, b), is perpendicular to the beam. The point J is dis- 
cussed in a later chapter. 


EXERCISES IV 


1. Find the values of x for which the following functions have 
maximum and minimum stationary values, discriminating the two 
cases 


1 $a? — Oe 4 ii, 4¢° — 1807 + 15a 4+ 7 
Te ee ee em LODO iv. 2° — 120% +) 459728 
v. w* — 242? + 64¢7 + 5 vi. — a4 + 62? — On? + 4a + 12 


vii. 324 + 8x — 6x? — 24” + 30 viii. a — 5a4 + 5a? - 1 

2. State the positions of the maximum and minimum stationary 
values of f(x) when 

Lei Ae) ee — 9 (ae es) ef (2) = (eae 

URE TA Cd Vat Sd (elie SON be (romeo Wd 

3. Discuss the stationary values of 

i. (w — 2)?(5 — 2a) ii. w?(x~ — 2) 
iii. w(x + 1) iv. (« — 1)%(a + 1)8 

4, Prove that the maximum and minimum values of a polynomial 

function occur alternately. 


Why is not this theorem true also for the case of the quotient of two 
polynomials ? 


5. Construct a cubic function of « which has a maximum value = 100, 
given by x = — 5, anda minimum = — 8, given by z = 1. 


6. Find the stationary values of the following functions, discriminat- 
ing between them 


ieee eteere ii, (1 — x)8/(1 — 22) 
Be 2 vt 
Gi eae te Pe ee) Moca ict 
l+a + 2 at — a4] 


7. Find the positions of the maximum and minimum stationary 
values of the following functions 


TERN hal lie ydeoeecn cena 1) iii. (x® + 10a) /(x? + 1) 


CHAP. IV EXERCISES 57 


8. Determine the positions of the maximum and minimum stationary 
values of 


jl. Sin % + cos x li, sin 2a + 2 sin x 
lil, sin 3% — 3 sin & iv. sin x cos 2x 
v. sin*x cos 2 vi. tan x tan (a — 2) 


9. Show that the minimum stationary value of 


sec @ cosec @ + Sec % + Cosec x 
ip 4°83...'. 
1 — 2a — 2 
10. Prove that ——______—, 
1+a — 22° 
always decreases as & increases. 


11. Find the stationary value of x2 + y when 2 and y are connected 
by the relation ay +y — be =1 Ans. 13. 


12. Prove that if y is obtained in terms of 2 from the equation 
wy(y — x) = 2a% 
the minimum value of y is 2a. 


13. A cubic function of x has values 4, 2, 6, 2 when xz = 0, l, 2, -—2 
respectively. Prove that the maximum stationary value is 6. 


14. A rod AB is to be divided at P so that AP? + 3BP? is a minimum. 
Show that AP: PB = 3:1. 


15. Show that the area of the greatest rectangle that can be inscribed 
in a semicircle with one side upon the bounding diameter is equal to 
the square on the radius. 


16. A point P is taken upon the arc of a quadrant of a circle whose 
bounding radii are OA, OB, and PM is drawn perpendicular to OA. 
Show that the area of the trapezium OMPB is greatest when the angle 
AOP is 30°. 


17. Show that the rectangle of greatest area that can be inscribed 
in a triangle ABC with one side lying upon BC has an altitude equal to 
half the altitude of ABC. 


18. Aline is drawn through a fixed point, whose Cartesian coordinates 
are (a, b), to meet Ox, Oy in A and B. Show that the minimum area of 
the triangle OAB is 2ab. 


19. A rectangular strip of paper ABCD, whose longer sides are AD, BC, 
is folded so that the corner A rests on BC at H, and the crease meets AB 
in P. Prove that the area of the triangle HPB is a maximum when BP 
is one-third of BA. 


20. Show that the isosceles triangle of greatest area with a given 
perimeter is equilateral. 


21. The lengths of the parallel sides of a trapezium are a, a + 22, 


and the other sides are equal and of length 6. Show that the maximum 
area is obtained by choosing 


a= t{+/(a? + 8b?) — a] 


58 INFINITESIMAL CALCULUS CHAP. IV 


22. Prove that the area of the smallest ellipse which can be drawn 
through the corners of a given rectangle is 1-57... times the area of the 
rectangle. 


23. By Parcel Post regulations the sum of the length and girth of a 
parcel may not exceed 6 ft. Prove that (i) the largest sphere that 
can be sent has 1°6,.. c. ft., (ii) the largest cube 1-7...-c. ft., (iii)»the 
largest rectangular box of square section 2c. ft., and (iv) the largest 
Cynder 2:6... Coit 


24. From the corners of a rectangular sheet of paper, 24 in. by 9 in., 
four equal squares are removed, and the sides then turned up so as to 
form a rectangular tray. Show that when the side of each of the 
Squares removed is 2 in., the tray has the greatest possible content. 


25. A symmetrical trough is constructed having its cross-section 
bounded by three equal lines of given length. Prove that its capacity 
is greatest when the section is half a regular hexagon. 


26. An open rectangular tank is to contain 288 c. feet, and the edges 
of its base are to be as 2:1. Show that the cost of lining it with lead 
is least when the depth is 4 feet. 


27. Show that the volume of the largest cone that can be inscribed 
in a sphere bears to the volume of the sphere the ratio of 8: 27. 


[If 6 is the semivertical angle and sin?0 = ¢, the volume of the cone 
varies as #1 — t)?.] 


28. Show that the volume of the largest cone that can be inscribed 
in a given cone with its vertex at the centre of the base of the given cone 
bears to the volume of the given cone the ratio of 4: 27. 


29. A conical tent is to have a certain content. Prove that the least 
amount of canvas is used when the semivertical angle is about 35° 16’. 


30. A right circular cone of given volume has the smallest possible 
total surface. Show that the area of the curved surface is three times 
the area of the base. 


31. A sector is cut out of a circular piece of paper, and the straight 
edges being joined a cone is formed. Show that the volume of this cone 
is a maximum when the angle of the sector removed is 2x(1 — 44/6) 
radians = 66° nearly. 


32. Show that the cylinder inscribed in a sphere whose curved 
surface is greatest has the curved surface equal in area to one-half the 
surface of the sphere. 


33. Show that in the cylinder, whose total surface is least for a given 
volume, the height is equal to the diameter of the base. 


34. A cylinder is inscribed in a cone whose vertical angle exceeds 
53° 8’. Show that the greatest value of the total surface of the cylinder 
is twice the area of the base of the cone, and that this is not a stationary 
value. 


35. The base of a statue 7 feet high is 9 feet above the observer’s eye. 
Show that the best view is obtained from a position 12 feet away. 


CHAP. IV EXERCISES 59 


36. Two ships, 4 and B, steam the one directly towards C and the other 
directly away, their speeds being as 2:1, and the angle ACB is 60°. 
Prove that when the ships are nearest, their distances from the port are 
in the ratio of 4: 5. 


37. A man in a boat three miles from A, the nearest point of a shore 
which is straight,‘wishes to reach a point on the shore which is 5 miles 
distant from 4; he can walk at 5 m.p.h. and row at 4m.p.h. Show 
that he should row 5 miles and walk one mile. 


38. Determine the most economical speed of a steamer which costs 
fa a day exclusive of the cost of coal, the expenditure of coal varying 
as the nth power of the speed (n> 1), and being £b per day when 
the speed is V. x 

Ans. Required speed = V E i 


(n—1)b 


39. Given that the force exerted by a circular electric current of 
radius @ on a magnet whose axis coincides with the axis of the coil 
varies as a(a2 + x2)? 


where z is the distance of the magnet from the centre of the circle, show 
that the force is greatest when x = 3a. 


40. The strength of a rectangular beam of breadth 6 and depth d is 
proportional to bd? and its stiffness to bd. Show (i) that in the strongest 
beam that can be cut from a cylindrical trunk the perpendicular from 
a corner of a cross-section on the opposite diagonal divides that diagonal 
in the ratio of 1: 2, and (ii) that in the stiffest beam the corresponding 
ratio is 1: 3. 


GTA dag Raa 
ALGEBRAIC FUNCTIONS 


48. Quadratic functions. 

The object of this chapter is to study f(x) by the help of its derived 
function f(z). The first functions considered are polynomial func- 
tions, which are of the type 


Aye” + ay +... + Oye + Ay 


We shall take for detailed study the simplest polynomials, and 
first consider the quadratic function, which we shall write 


f(x) = ax? + 2bx + 
its derived function being 
f'(@) =2ax + 2b = 2(ax + b) 


The graph of f(x) is a line whose slope to a person describing 
it in the direction of x increasing is up if a is positive, and down if 
a is negative. The node of f(x) is given by x = — b/a; it is an 
ascending node if a is positive, and a descending node if a is negative. 
Hence f(x) has a stationary value f( — b/a), which is a maximum if 
a is negative, and a minimum if a is positive. Again, 


f(- b/a) = ab?/a? + 2b(- b/a) + 
= (ac — b*)/a 


The quantity (ac — 6?) is called the discriminant, and is indicated 
by A. The stationary value of the function is A/a. 


First, let us take the case in which A 1s positive. If a is positive, 
the stationary value is positive and is a minimum. Hence the 
function is always positive. If a is negative, the stationary value 
is negative and is a maximum. Hence the function is always nega- 
tive. In neither case does the graph cross Ox. Therefore when 
A is positive, the equation f(x) has no real roots. 


Secondly, let us take the case in which A is negative. If a is 
positive, the stationary value is negative and is a minimum ; in this 
case the graph cuts Ox in two points, one on each side of x = — b/a. 


CHAP. V CUBIC FUNCTION 61 


If a is negative, the stationary value is positive and is a maximum, 
and, as before, the graph cuts Ox in two points. 

In both cases the equation has two roots which are equidistant 
from x = — b/a. 


Thirdly, when A = 0. In this case the stationary value is zero, 
and the graph of f(x) touches Ox at the point where the graph of f’(x) 
crosses it ; the graph of f(x) is above Ox if a is positive, and below 
if a is negative. The equation f(x) = 0 has equal roots, namely 
w= — b/d. 


The character of the function is entirely determined by the signs 
of Aanda. This result also follows by writing the equation in the 
standard form of Art. 9, 


f(x) = a(x + b/a)? + (ac — b*)/a 
= ax + Aja 

where xv’ = x + b/a. This form of the function in which w’ is the 
new independent variable corresponds in the graph to a change of 
origin, the new origin from which 2’ is measured being at a distance 
b/a to the left of O. If O’ is the new origin, the new axis of y is the 
axis of symmetry of the function ; it is spoken of as the axis of the 
curve, that is, the axis of the parabola which is the graph of the 
quadratic function represented. 


49. Cubic functions. 

By shifting the position of the origin, the general cubic function 
may be reduced so that the term involving x? is wanting. This 
point will be illustrated sufficiently by examples. We shall also, 
for the sake of simplicity, consider only cubic functions in which the 
coefficient of x? is unity. The specialised cubic which we shall 
discuss is of the form 

f(z) = xz? —- 3px + q 


whence f(x) = 3(a — p) 


Now, if p is negative the graph of f’(x) does not cut Oz; hence, 
unless p is zero or positive, the function has no stationary values. 
Taking p as positive, the stationary value corresponding to 


paay 2S SVP) = pvp — 38pV/p +7 = 4 - 2pv/p 
Similarly f(- VP) = 4 + 2pr/p 


Writing A = f(v/p).f(- Vp) = @ —- 4p3, we have a quantity 
which is of importance in discriminating the nature of the cubic 
function. For, if A is positive, f(,/p) and f(-— +/p) are ether 
both positive or both negative ; while, if A is negative, one of the 
stationary values is negative and the other positive. Again, 
when p is negative, although there are no stationary values, yet 
A exists, and in this case, as p? is negative, A is always positive. 


62 INFINITESIMAL CALCULUS CHAP. V 


We can now discuss the nature of the cubic function according 
as A is positive, negative or zero. 


First, when A is positive. 
(i) Let p be positive ; there are then two stationary values, and 
they are both positive or both negative. The graph of the function 


(a) Fie. 15. (b) 


has one of two forms sketched in Fig. 15. It is clear that the 
function vanishes either for a value of x < — 4/p, the value which 
gives the first stationary value arrived at as the curve is described 
in the direction of the arrow (Fig. 15, a), or for a value of x which is 
greater than 4/p (Fig. 15, 6). 


(11) Let p be negative ; there is then no stationary value. Since 
f’(x) is positive for all values of x, the function is always increasing, 
and its graph crosses Ox only once. If 
q = f(0) is positive, the crossing point is 
to the left of O; if qg is negative, it is to 
the right ; the sign of the single real root 
of f(x) = 0 is opposite to that of q. 


Hence, when A is positive the cubic 
equation f(x) = 0 has one and only one 
real root. 


Secondly, when A is negative. 

In this case p is positive, for g? — 4p? is 
negative ; and the stationary values are 
one positive and one negative. As in the 
previous case, x = — +/p is a descending node of f’(x) and x = +/p 
is an ascending node. Therefore, f( —+/p) is a maximum and f(1/p) 
is a minimum, Again, from the values of 


f(— +/p) and f(+/p) it is obvious that 


f- Vp) > f/) 

therefore f(-— +/p) is the positive stationary 
value and f(./p) the negative one. The 
diagram is shown in Fig. 17. In this case the 
graph of f(z) must cross the axis three times, 
and the equation f(z) = 0 has three real roots, which are separated 
by the numbers +/p, — «/p. In other words, the roots of f(x) = 0 
are separated by those of f’(x) = 0. This theorem is true for any 
polynomial function, and is called Rolle’s Theorem (see Art. 151). 


Fic. 16. 


Fico iv. 


CHAP. V INFINITIES 63 


Thirdly, when A = 0. 

In this case, either the maximum or the minimum stationary 
value vanishes. Now, if q is positive, f(— +/p) cannot vanish. 
Therefore f(4/p) must be zero ; while, if g is negative, f(- +/p) = 0. 


(a) Fie,18. ° 


The two cases are illustrated by the left and right-hand diagrams 
of Fig. 18. The result is confirmed by considering the sign of 
f0) =4¢. 

Again, if A = 0, q? = 4p, and if ¢ is positive 


f(x) = «3 — 3px + 2pr/p 
(x — +/p)(x + 24/p) 


while if g is negative 
f(a) = x — 3pu — 2pr/p 

(a + ~/p)*(e@ — 2r/p) 

These resolutions enable us to solve the equation f(x) = 0 com- 

pletely. 


50. The infinities of linear, quadratic and cubic functions. 


As x increases beyond a certain value, the graph of a polynomial 
function passes beyond the limits of the paper, and it is usual to 
speak of the graph as proceeding to infinity. Now the ultimate 
parts of these curves are disposed with respect to the axes in one 
of four ways, which may be described by the quadrants in which 
the graph ultimately lies ; the four alternative dispositions are 


(A) in the second and first quadrants 

(B) in the second and fourth quadrants 

(C) in the third and first quadrants 

(D) in the third and fourth quadrants 
The infinite branches of the linear and cubic function are of the 
B or C type, and this is true of all odd polynomials ; while the 
quadratic and other even polynomials are of the A or D type. 

The above statements are inferences drawn from the graphs. 

The same results may be deduced by substituting large values of x, 
positive and negative, in such pairs of adjacent functions as 


—-2r4 +3 Sr 
ge? —3¢4 +3 x 
-2+2-3 ise 


64 INFINITESIMAL CALCULUS CHAP. V 


in which the second member of each pair consists of the term con- 
taining the highest power of xin the first member. If in these three 
pairs we write 2 = 100, we obtain 


— 197, — 200; 9803, 10000; - 999903, — 1000000 


values which are so nearly equal that it would be impossible without 
great labour to distinguish between them ona graph. The relative * 
differences in the various cases are all small; thus, neglecting signs, 
we write them as 


8 es O-D1Geh ieee Tet en) OT ee nee OC an 


We could, however, by taking x large enough, make any one of the 
relative differences as small as we please. To show how much 
smaller the relative differences become when # is taken large, we may 
select x = 108 in the linear function, when we obtain a relative 
difference = 210-*, 


We will now prove that the relative error made in taking ax® instead 
of ax? + bu? + cx + d can be made as small as we please by taking x 
sufficiently large. 


The relative error is 


= (bu? + cx + d)/ax? 
= (b/a)a-! + (c/a)u* + (d/a)x- 


If p is the greatest of the coefficients b/a, c/a, d/a, without regard 
to sign, the relative error is not greater than 


Dover eee ao) 


which again is less than 3pa-!. If ¢ is the standard value below 
which it is desired that our relative error should fall, the required 
result is obtained by taking x = 3p/e. 


We have proved that if, for instance, an error of 1 in 1000 is 
negligible, a value of x can be assigned such that, for it and for 
greater values of x, no distinction need be made between the graphs 
of az? + ba? + cx +d and that of az’. We conclude that when 
a — , the graph of the cubic function is in the first quadrant if 
a is positive, and in the third when x —~ -— o, while, in the case of 
a negative, the graph is in the second and fourth quadrants when x 
is very large. 


51. Illustrative examples. 

A few examples are given to show the student how important it 
is to associate the study of the derived function with that of the 
function. 


* The relative, or proportional, difference between .m and n is 
(m — n)/m or (m — n)/n 


CHAP. V EXAMPLES 65 


Vib oad a f(x) =-2® — 3a? - 92 + 7 
F(a) = 3(2* — 22 — 3) = 3(@ — 3)(x + 1) 


The stationary values of f(z) are given by x =— 1,3. Again,w =—- 1 
is a descending node of f(x); hence f(— 1) = 12 is a maximum, while 
(3) = — 20 is a minimum. The equation has three roots separated 
by the numbers — 1, 3. 


iste es, Saje= 457) — 1207 + 127. — 2 
f'(%) = 12(@? — 2% + 1) = 12(% - 1) 


The stationary value of f(x) is given by x = 1, but as /f’(a) does not 
change sign, the stationary value is not a maximum or a minimum. 
As f(z) is never negative, f(x) never decreases as x increases. Again, 
since f(0) = — 2, the equation has one positive root, and this root lies 
between 0 and I, since f(1) = 2 (see Fig. 19). 


Fic. 19. Fic. 20. 
Ex. 3. f(a) = O-lat — 1-4a? + 24a — 09 
f(x) = 0°-4(~ — 1)(a@ — 2)(a% + 3) 
The stationary values are f(— 3) = — 12:6, a minimum; /(1) = 0-2, 
a maximum; and /(2) = — 0-1,a minimum. The curve crosses the 


axis when x= — 4°5, 0:6, 1-6, 2°3 nearly, and is drawn in Fig. 20. 


52. Rational algebraic functions. 


The rational integral algebraic function, or the polynomial of 
degree n, is here denoted by P,,(~) ; thus, the linear function is P,(a), 
the quadratic is P,(w) and the cubic P,(x). The quotient of two such 
polynomials will be denoted by R(x); it is the rational algebraic 
function, thus P (2) 


P,,(@) 


R(x) 


C.C, E 


66 INFINITESIMAL CALCULUS CHAP. V 


We shall suppose that P,,(~) and P,(x) have no common 
factor.* 

In drawing the graph of R(w), there are certain parts of the curve 
to which we must pay particular attention, if we wish to obtain a 
general view of its form, and therefore of the general character of the 
changes which the function undergoes in its range, or ranges, of 
definition. These parts are 

(1) points at which the graph of R(x) crosses the axis of x, which 
are given by the roots of P,,,(x) = 0, these are the zeroes of R(q) ; 

(2) points at which the graph of R(w) becomes infinite, which are 
given by the roots of P,,(x) = 0, these are the infinities of R(z) ; 

(3) points at which R(«) has stationary values, given by the roots 
Of (a) =105 

(4) the form of the graph as the two ends of the continuum are 
approached. 


53. Reciprocal of the polynomial function. 


Before discussing in detail the general function R(«), we shall 
illustrate our methods by the simple though important case of the 


BiG. 21 


reciprocal of the polynomial function. The case arises when m = 0; 
we write — R@) =1f@) B@) =- f@fwr 

The function R(x) has no zeroes, its infinities correspond to the 
zeroes of f(x), and its stationary points are given by the same values 


* Tf P,.(x), P,(a) have a common factor f(x), then R(x) is undefined for values of 
x which are roots of f(a) = 0, but does not differ otherwise from the function obtained 
by clearing the numerator and denominator of this common factor. 


cHaAP.v RATIONAL ALGEBRAIC FUNCTION 67 


of # as those which give the stationary values of f(x), namely, by 
the roots of f’(x) = 0. Since R’(#) and f’(x) have opposite signs, 
the maximum stationary values of f(x) correspond to minima of 
R(x) and the minima of f(z) correspond to maxima of R(x). When 
R(x) increases, f(x) decreases ; indeed, the graph of R(x) is most 
readily constructed by first drawing the graph of f(x). In the’ 
diagram (Fig. 21) ABC is the graph of f(z). 

We note that R(w) and f(x) have the same sign, and therefore the 
graphs of the two functions are both above or both below Oz for 
the same x; also, as f(x) crosses Ox descending as at B, R(x) has an 
infinite discontinuity, changing from # to — o. At C, however, 
R(x) — -— » on both sides of C, f(x) vanishing but not becoming 
positive. To the maximum of f(x), which lies between A and B, 
corresponds a minimum of R(#), and corresponding to the minimum 
between B and C there is a maximum of R(x). Other important 
points are those at which the graphs cut, here 


f(z) = R(x) = I/f(a) 
therefore jit sae ag Nl 


Again, as xo, and as x—~ — », R(x) —0, the position of 
the graph of R(x) being decided by the fact that R(x) and f(z) lie 
on the same side of Oz for the same value of a. 


54. The general type R(x) = P,,(x)/P,(x) 

A few general hints may be given as to methods which are appli- 
cable in drawing the graph of R(x), but it is impossible to deal here 
with all the numerous cases which may arise. The student may 
learn much from the worked-out examples which follow. The 
questions raised in Art. 52 must be first dealt with, and the 
zeroes, infinities and stationary values of the function determined. 
The discussion of the behaviour of R(#) when 7 ~ — o and when 
2 —~o will be treated more fully by help of the principles laid 
down in Art. 50. The relative error in taking av” for P,,,(x) and 
a’x” for P(x) were shown to be less than any assigned value, if 
suitably large values of ~ were taken. If x exceeds the larger of 
these values, we may study the graph of R(x) at the extremities 


of its range by examinin 
ge by g ax” 


an 
The’ discussion now varies according as mis >, = or <n; these 
three cases will be considered separately. 


I. If m <n, we substitute for R(x), when 2 is very large, the 


expression P 


a’xyn-m 


68 INFINITESIMAL CALCULUS CHAP. V 


Now, this expression is for large values of 2 a small quantity which, 
if m — m is even, has the same sign as a/a’ whether 2 is positive or 
negative. The graph of R(x) in this case approaches Oz, lying on 
the same side of it at both ends. 

But, if » — m is odd, R(w) lies on opposite sides of Ow at its 
- extremities, being above the axis when w — if a/a’ is positive, and 
below if a/a’ is negative. 


Il. If m =n, R(x) > a/a’, when x—o and when ~~ —- ow. 
Here the position of the line y = a/a’ with regard to the curve is 
similar to that of Ox in Case I. It is usual to express the geome- 
trical fact by saying that y = a/a’ is an asymptote of the graph. To 
settle the side upon which the graph lies requires the expansion 
of the function R(x) in a series of powers of 1/x. This process is 
explained below in connection with some examples. 


Ill. If m> n, we divide P,,,(x) by P,,(x), and write 


where f(x) is the quotient and P,,_,(x) the remainder. 
Let Q, and Q be points on y = R(x), y = f(x) respectively having 
the same abscissa which is large in magnitude. Then 


QQ, = Ria) — fe) = Pr_s(@)/Pr@) 


By the method used above we can take x so great that the right- 
hand side is, to our degree of approximation, 
ie 


aa” ax 


where a,2"—! is the term of highest degree of the remainder. 

The signs of a,/a’ and x determine the relative position of the 
graphs of R(x) and f(x). Itisclear that when vis very large the error 
made in taking f(x) for R(x) is negligible. The curve y = f(x) is an 
asymptotic curve, and when f(a) is linear it is called an oblique asymp- 
tote. In certain cases it may happen that the remainder is of the 
(n — 2)th degree; in this case, the approximate value of QQ, is 
a,/(a’'x?), a quantity which does not change its sign with x; here, 
R(x) is above or below f(x) at both ends. 


55. Illustrative examples. 


x —) F 2+ 2a — a 
in eg 1 R(x) CeO R’(x) Timea 


The graph crosses Oz at x = 1; its infinities are given by a = 0, 
x = — 2, and its stationary points are the roots of | 


u*-—2%-—-2=0 


CHAP. V EXAMPLES 69 
namely, 2 = dh 4/3 
R(1 + 4/3) = #(2 — +/3)= 0-13... and R(1 — 4/3) = 1°87... 


When 2 is very large, the value of R(x) may be calculated from its 
approximation 1/x; this shows that the graph of R(w) as ~~ - @ 
and as x--~« may be found by drawing the corresponding portions 
of the rectangular hyperbola y = 1/2. 

In the range [— ©, — 2] R(x) is negative, in [ — 2, 0] it is positive, 
in [0, 1] it is negative and in [1, © ] it is positive. 

The above facts are summarised in the firm lines of the diagram, 
Fig. 22; the student will find no difficulty in filling in the dotted parts. 
It is unnecessary to examine in detail these parts except for some par- 
ticular purpose. The student should aim at acquiring the power of 
sketching rapidly the graph of such a function as this. 


Se 


RIGmoce Fic. 23. 
Ex, 2 aygjy <a ne ag 
ia Ce os. a xc— 2 
; 5 


The graph of R(x) is a hyperbola, Fig. 23, which does not cross Oz ; 
at x = 2, the function is infinite and undefined ; its left and right limits 
as x increases through this value are respectively — © and o. Again 

y=u+2 


is an oblique asymptote. When 2 -—- 0, R(x) is above the asymptote ; 
at the other end, below it. The stationary values are given by 


~ = 2+ 4/5, and are equal to 4 + 2/5, that is, to 8-47..., — 0-47... 
fs x(a — 1) ve ee, 2(1 — 2a) 
Pisce ose ret a R(z) = @ = ae 4 2 


(x — 2)(a + 1) 


70 INFINITESIMAL CALCULUS CHAP. V 


ihe graph crosses Ox at a= 0) a = 1352 '— 4 givesta mania 


value = §. The lines 2 = 2, x = — 1 are vertical asymptotes. There 
is also a horizontal asymptote y = 1 deduced from 3 
2 2 
R = — a= | 
(2) 1 + Rael aap 1+ 32? nearly 


The asymptote is below the curve at both ends, Fig, 24. 


AG 0 x 
Fie. 24. Fic, 25. 
1 (1 — x 
Ex. 4. Be = SOBA ABU en R(x) = - ( x") 


a —x¢ +1 (a2 — 2 + 1)? 


The curve y = R(x) does not cross Ox, and has no vertical or oblique 
asymptote. The maximum value is given by R(1) = 3, and the 
minimum R(- 1) = 4. Again, x being large, 

22 


pia mr teak o* 5 Spee 


2 
Shee = 
+ iS nearly 


It follows that y = 1 is a horizontal asymptote, which is below the 
curve when «-—- © and above it at the other end. It may also be of 
help to notice that R(0) = 1, R’(0) = 2, see Fig. 25. 


56. The function given by y* = R(x). 


The most general type of algebraic function is given when x and y 
are united by the relation 


yf) + y fila) +... + Ufnalm) + faz) = 0 - 


where f(x), f,(@) ... f,(x) are polynomials of any degree. Functions 
of this kind will occur often at different stages of the subject, but 
at present the case which we select for discussion is the algebraic 
function, which is defined by 


y? = Riz) = P,, (a) /Pp(2) 


We first draw the graph of R(x), and then notice that y is not defined 
when R(x) is negative, while when R(z) is positive there are two 


CHAP. V ALGEBRAIC FUNCTION 71 


values of y equal in magnitude and opposite in sign. Other impor- 
tant, though simple, facts are expressed analytically by 


Rw) =0 y=0; R@w+~+0 Rf re 

An interesting feature of the graph is the point at which R(x) =0; 
three typical cases are expressed by the equations 

y=a(ex-aRw y=(e -—a)R (x) y*® = (x — a)?R,(2) 
where R,(a) + 0. In the first case the student may prove that the 
gradient at 2 = a@ is infinite; that in the second it is + W/R,(@); 
that in the third it is zero. For further illustration the reader is 
referred to Ex. 1, 2, 3 which follow. 


57. Illustrative examples, 

Hoe al y2 = a(@ — 1)\(v — 2) (Fig. 26) 
Here, in the ranges[ — « , 0] and [1, 2], y is undefined ; in (0, 1) the values 
of y give an oval, while in (2, © ] we have a symmetrical branch extending 
to infinity. The infinite branch is best realised by considering the semi- 
cubical parabola y? = «* to which the curve approximates when z is large. 


O XC 
Fic. 26. PIGS 2a. 
rox 2, y2 = (x — 1)\(@ — 2)% (Fig. 27) 
Here, when x < 1, y is undefined. Also 
dy x — 2 
= ey eee 2 Ae ihe 
HE. ORE aE VAG ae 


Hence, at (2, 0) the gradient is equal to + 1, and the curve crosses 
itself at this point. Ataz = 13, y has stationary 
values. 

| Sip eas § y2 = (x — 2)? (Fig. 28) 
In this curve, y is defined only when x= 2; 


the point (2, 0) is a cusp, the branches touching 
Ox at this point, for 

dy 3 1 

Cee oe OVA 

dx 3 ) 


* This is an anticipation of results proved in Chapter VI. 


Fic. 28. 


72 INFINITESIMAL CALCULUS CHAP. V 


x+i1 : 
1 ear: © a Fig. 29 
y eS (Fig. 29) 


A consideration of the function (w + 1)/(% + 2) shows that when 
x—+> ow, y*-— 1, that y* is positive, except in the range(— 2, — 1). The 
curve required has two horizontal asymptotes y = + 1, and a vertical 
asymptote x = — 2. 


Fic. 29. Fic. 30. 


(2 + 1)? 
EXD: oto Se Se Fig. 30 
x y eae (Fig. 30) 


The graph of (w + 1)#/(x + 2) is a hyperbola whose asymptotes are 
x = — 2,y = «x, and which touches Ox at x = — 1; also this hyperbola 
is above Ox from [ — 2, 0]. The curve whose equation is given may 
be drawn from these facts about the hyperbola. It can be shown that 
the gradient of the given curveat (— 1,0)is + 1. Its infinite branches, 
which are in the first and fourth quadrants, are given 


by the approximation y” = x. —— 
ye “oe al 


Ex. 6. Be RSA, Fig. 31. 


The range of values of x for which y is defined is ( — 1,  ], stationary 
values are given by x = 0, for we have, as will be proved in Chapter VI., 


EXERCISES V. 


1. Express the following quadratic functions as the sum, or the 
difference, of the square of a linear function and a constant 
i, 380% — 64 + 1 ii. (wv + 3)? + (x — 5)? 
iii. 2(24 + 1)% — (a — 3)? iv. x — (w% + 1)? — (a + 2) 
v. x2 — (w« + 1)% + (a + 2) vi. a? + (w@ + 1)? — (a + 2/7 
2. Trace the graphs of the following cubic functions 
io — 374+ 1 ii. a — 384 + 6 iii, a? + 64 — 9 
iv. 2 + 6x + 1 v. 2 + 302 — 92 vi. a3 + 32% -— 9x + 6 
vii. 2? + 32% — 9a — 28 viii. a® + 207 — 5a ix. 2° + a* — 64 4+ 3 


CHAP. V EXERCISES 73 


3. Trace the graphs of the following biquadratic (or quartic) 
functions 


1 a® —' De= 2 ii. wt — Qa? + 3 
ill, 7? — 4a? + 497 + 1 iv. zt — 4a + 4a? -— ] 
v. dat + 423 + 62? — 2 vi. 304 + 423 + 622 + 2 
vii. zt — 622 + 8x — 3 viii. v* — 62? + 8a + 25 


4. Prove that x* + aa + b can always be resolved into the product 
of two quadratic factors with real coefficients. 
Resolve into real quadratic factors : 
i att a?+il li. vt — 3o¢?2 + 1 iii. 24 + 9 


5. Show that 2t — 4x + 10 = 0 has no real roots, and in general 
that «4 — 493% + gq = 0 has no real roots, if g > 3p. 


6. If f(z) = 0 has a single root in the neighbourhood of # =a, then 
a, =a - fa/fa) 


is nearer the root than a, if for all values of w in [4(a@ + a,), a] the 
relation | (f’(x)|= 2| f’(a)| holds. 

Note.—If the tangent at the point (a, f(a) ) of the graph of f(x) meets 
the axis of z at T and if NP is the ordinate of P and A bisects TN, then 
x = OT isa better approximation to the root of f(z) = 0 thang = ON, 
provided the point Q, where the graph cuts Oz, isin 7'A ; Q is certainly 
in 7'A, if the gradient of the graph at all points above AN is less than 
the gradient of AP. 


7. Find approximations to the roots of the following cubic equations, 
using Question 6 
i. w® — 9x — 14 = 0 li, 2° — 127 + 20 = 0 
iii. vw? + 37 — 244 — 16 = 0 
8. Prove that the real roots of at — 12% — 5 = 0 are approximately 
Bale Od. 


9. Draw the graphs of the following functions, determining any 
stationary values they may have 


a ees ea | .., w(@ — 4) 
he aa el i Pata bin Secs 
Liat aaeteia a x? — 4 x2 +9 
eo 19 x* + 8% + 16 . a + Ie — 6 
iii7) ieee View Vie 
a(x — 4) x — 64 +9 z+ 5 
2 a 
Sfp ta A oof, TEE SANs RU Sc eneatcaaee Ch 
v2 4+ 2a + 9 (w — 4)(~ — 5) (c — 1)(a — 3) 
2 2 
a (* — 1)(@ — 5) ty (ax + b) Pte atta 24 + 3 
(x — 2) x 3a" + Yo + 1 
10. Show that the stationary values of 
a + a? 
(~ — c)(x — d) 


are given by rational values of a, if (a2 + c?)(a* + d?) isa perfect square. 


74 INFINITESIMAL CALCULUS CHAP. V 


(aw — b)(bu + a) 


ll. Prove that wns jt 
(bu — a)(aw + b) 


has all values. 
ax? + be +c 
cx? + ba +a 


has no stationary values if b? > (a + c)?.. What is the type of the graph 
of the function when b? < ac ? 


12. Show that 


13. Sketch the curves represented by the following equations 


e a ey ans 
1g? ea 4) HS pl serse iY? aa 
: x +2 re | w+ 1 
ivi 42 Se Lee en yes 
A oo if es Tiae on | 
vil ape ville eh ee ix if ae 
nee Nie nie hae tent Fae 
x. y? oe ee, X1 y" SS ee — 1)(x — 2) 
CAT ee y 
*s ee 1 — 2 1 + a 
xii, y? = a3 (a — 2) Xi. 42 = oa Tae SUV ane ig 


Cineke Ibi Sa ke Aa 
INVERSE OF A FUNCTION, FUNCTION OF A FUNCTION 


58. The square root function. 


There are certain mathematical operations which are called direct, 
such as addition, multiplication and involution. We have been 
concerned hitherto mainly with functions which arise from the use 
of such operations, types of which are 


x +2 3x es 


But to each of the three direct operations mentioned above, an 
inverse operation corresponds. Passing over the operations of sub- 
traction and division, which are respectively the inverses of addition 
and multiplication, we base the explanation of inverse functions upon 
the inverse function associated with the simple quadratic function 2. 
We have used the table 


Meese eto | a | ee 


Bs cee inate ee ane a Ue ad 


to illustrate the part played by a mathematical operation in the 
construction of a function. It is necessary to remind the student 
that the selection of the values of x is purely arbitrary and is made 
from a vast assemblage of numbers, rational and irrational, any of 
which might have been taken—the whole of which constitutes the 
arithmetical continuum. The selection made is a concession to 
human weakness and its preference for the integral elements of the 
continuum. Thus, the table might have been written 


x |—2|—1-738 ~1-414| — 1) 0 | 1 | 1-414 L732, D 2-236 


x| 4 | 3.00 2-00 | 1 | 0 | 1 | 200, 3.00 | 4 5.00 


In this form the selection of values of the independent variable 
gives values of the dependent variable which to two places of 


76 INFINITESIMAL CALCULUS CHAP, VI 


decimals are integers. Now suppose that we interchange the first 
and second row, calling the new upper row x and the new lower row 
f(x); we then have the table of a new function, which may be written 


0 ee 


fi)| 0 | +1 +144... Sippy eee | + 2-236... 


The new function is closely related to the old function of squares. 
What label can we apply to the new function? The answer is 
supplied by our knowledge of arithmetic ; the new function is the 
square root function, the function which we denote by the symbols 


+4/x or #?; the numerical magnitudes in the lower row being 
derived from the corresponding terms of the upper row by taking 
the square root. 


59, General inverse functions. 


The process which we have followed in deducing the function 2? 
from the function 2? is perfectly general. For the correspondence 
between two sets of numbers, which we term the independent (qa) 
and the dependent variable (y),implies also a correspondence between 
the y’s and the 2’s, in which y plays the part of the independent 
variable. Hvery function is therefore associated with a second function 
called its inverse, the same pairs of values being cowpled in the tables 
of the functions. 


60. Some properties of inverse functions. 


The polynomial functions are single-valued, that is, to each value 
of x a single value of y corresponds ; also their range is unrestricted, 
that is, whatever number is assigned to a in the arithmetical con- 
tinuum, a corresponding value of y can be found. But the inverse 
functions of single-valued functions of unrestricted range need not 
be single-valued, and may be restricted in range. In the case selected 


Are . . rs 
x? is a function which has two values for every positive value of x, 
no value when z is negative and a single value when 2 = 0. 


The student should form by the help of tables the functions which 


are the inverses of e +2 By a3 
and he will find that he obtains the functions 
a — 2 dx a 


These three functions are, like the original function, single-valued 
and of unrestricted range. 


61. The graph of an inverse function. 


The connection between the graphs of a function and of its inverse 
can be studied best by drawing the two graphs upon one diagram, 


CHAP. VI INVERSE FUNCTION 77 


Let (x, y) be P, a point on the graph of the function, and (X, Y) 
be P’, the corresponding point on the graph of the inverse function. 


Then xy Ves 


It is obvious (see Fig. 32) that the perpendicular from O upon 
PP. bisects PP and also bisects. the angle Oy. Then P, P’ are 
the images of each other in the diagonal line 
y =x of the square paper which passes 
through O, and this is true for each pair 
of corresponding points. It follows that the 
graph of a function and of the inverse func- 
tion are the rmages of each other in the diagonal 
Tae 

It follows from either the arithmetical or the 
geometrical discussion that the inverse of the 
inverse of a function is the original function. 

Certain functions are their own inverses, e.g. the functions — 2, 
1/x. The graphs of such functions are symmetrical with respect 
to y = @. 


Fie. 32. 


62. Differential coefficient of an inverse function. 
It is required to prove that 


dy dx 

dx dy 
We begin with y = f(«#), and represent the inverse relation between 
the same pair of variables by x = o(y), or, if in the second relation 
we put x = Y, y = X, we have Y = o(X). 
The relation which we now have to establish 

takes the form 


F(a) .¢(X) = 1, 
and will be proved first from the geo- 
metrical properties of the graphs of f(z) 
and o(X). 

The tangents at P and P’ being sym- 
metrical with respect to the diagonal line through O must meet on 
cc PTx = P'T'y = 90° — P'tx 
provided neither tangent is parallel to an axis. It follows that 


fate re cot Pita 
f'(@) = 1/¢"(X) 
f(a .e(X%) =1 

dy dx 


that is, tid | pats, WT | 
dx dy 


Fic. 33. 


78 INFINITESIMAL CALCULUS CHAP. VI 


This method of proof brings out the fact that in dy/dx* y is 
thought of as a function of x, while in dx/dy x is a function of y, 
namely, that function which is the inverse of the first function 
considered. 

In the following analytical proof this important consideration 
may be easily overlooked. 

Let x, y be a pair of corresponding values of the function; y, x 
are the corresponding pair of the inverse function. Further, 
x + 6x, y + dy is a consecutive pair of corresponding values ; then, 
since 6%, dy are both finite, 

Oy Ox _ 
du Oy 
Now as 0x — 0, dy — 0, and 
as 0x > 0, wee = dy 7, (Where x is independent variable) 
On Oe 
as Oy +0, ay - a (where y is independent variable) 

With the assumption that neither of these limiting values vanishes, 
we have the theorem to be proved 

dy dx hh 
dx * dy 
63. The differential coefficient of 2/x. 

BE Te ierntas x", where q is an integer, then the inverse function 

is given by the relation a= yf 


This function we can differentiate by the rules laid down in 
Art. 37, and we have 


dx 
—— = Qa) az gaka- Dia 
dy Vy qu 
Hence dy = ] dat = Lae = L aq-1 
da dy 4 q 


64. The differential coefficient of x”. 
First, we take p, q as positive integers. Then 


Yi erie = ele Ae). p. dactore) 


ee (Yar © gx +... +... (p terms) 


= px - Dia oD ae-1 : Deltas 
q 
*This notation for the differential coefficient is used very sparingly in this book 


and then only for typographical reasons. It does not imply that the differential 
coefficient is a fraction with a numerator and a denominator. 


CHAP. VI COMPOUND FUNCTION 79 


Secondly, we take the case in which 


d 
pla — —_ pla 
dy Ova aa 
dx Pld 


— — -— Pid — x2v/a 
dx 


LIN EES RIF CR Se per 


q q 


The results proved allow us now to assert that for all com- 
mensurable values of 1, 


dx 


The discussion of incommensurable numbers which has been given 
may perhaps prepare the student to accept the extension of this 
formula to the case of m being an incommensurable or irrational 
number, the knowledge which he possesses is hardly sufficient to 
justify us in embarking upon a formal proof of the theorem. 


65. Function of a function, compound function. 

The function 4/(%? + 1) is chosen to illustrate the class of functions 
considered under this heading. The construction of such a function 
depends upon two tables, a table of squares and a table of square 
roots. The function is the square root of a quadratic function 
of x; its structure may be exhibited by writing down the 
following sequence of functions, each of which specifies a stage in 


the construction " 


re 
v2 +] 
a/ (a? + 1) 


Two intermediate stages are shown ; the process might have been 
shortened by the omission of the first stage, in which case we should 
pass from x to the quadratic function 7? +1. This process is 
expressed by an analysis in which zis substituted for the intermediate 
function. Thus 


aT raed | piesa (toa man a 


The reader may notice that the importance of this analysis for us 
lies in the fact that we have resolved the function into two, each of 
which can be differentiated by the rules already laid down. Thus 


80 INFINITESIMAL CALCULUS CHAP. VI 


It is with this object that we break up the compound function into 
its simpler elements, the intermediate functions used being those 
whose differential coefficients we can form. 


66. The differential coefficient of f[¢(x)]. 

Let us write y = f(z), 2 = o(x), where we suppose that f’(z), 9’(x) 
are known, and by their nature are finite. Let x be changed into 
a + 6x, and let the value of z (which is paired with 2 in the function 
z = o(x)) become z + 62; further, let y + dy correspond to the 
value z + dz. Then we have 

y + oy = f(z + 62) z+ 6z = o(@ + dz). 
Now, since 6x, dy, dz are all finite increments, 
dy _ dy oz 
Of OO 
And as 6% — 0, we have dz + 0, and consequently dy > 0. 


Again, as 6x > 0, 
dy dy dy dz dy dz 
dx dx 62 du dz dx 
and since in all stages of the sequences which these limits imply 
PEI REE 
0% B02 Hox 
the limiting values of the two sides are equal, that is, 
dy _dy dz 
kT PA fe 
The same result may be written by means of the functional symbols 


y = fle@] 


in the form J = f’[(x)]. 0'(x) 


Taking the example given in Art. 65, 
yarvV(e+ 1) =2 neitiigt a th 
1 dz 


dy 1 -4_1 aoe 
Pee a OPV AE nities Weld 
dy dydz 1 1 1 
Theref OY sg ROY Ue oa lie haa 9) a 
The Arapleitde i da: GoaroAy Cea] GV OR CD) 


The method given above is of great importance, as it allows us 
to increase very considerably the stock of functions which we may 
differentiate. In some of the examples worked out below, alterna- 
tive methods of differentiation are employed in order to bring the 


CHAP. VI EXAMPLES 81 


newer method into relation with the processes explained and used 
at an earlier stage, but it is not always possible to give alternative 
solutions. 


67. Illustrative examples. 


eee y=(1 —2 =1 — 3x.4+ 32? — 23 
dy _ : aa hare 2 
mee Ore oe 3 We) 
dy ~ d(l — x)® -d(1 — z) 
Again, hl es, Maes ieee he a\th 
See 3 dx d(l — a) dx See sagt) 
= —3(1 — x) 


The same result is obtained ; the second method may also be carried 
out with the substitution z = (1 — 2). 


a2 vel 2 
Es: y = (; y= 
1+ 1 + 2u + x? 
dy  (-— 2 + 2x)(1 + x)? — (2 + 2a)\(1 — x? 
dx (1 + x) 
Pty 4(1 — a) 
7 (1 + x) 
Also, writing z = (1 — 2)/(1 + ~), y = 2 
CE, CURES 5 neh 0 ied Nee Cad am Ppt Ls 
Cree. Oa (1 + x) (ieee )2 
oes la 2) 
(ORY 
Ex. 3. y= WG a5 *) 
| ne 
1 1+2 
Pete Vie z= 
1-2 
Ae RAN ese enes gal i) dz 2 
deh mio cale 28 a dceme (ib a)2 
dy dydz _ 1 
de dzdx (1 —x)4/(1 — 2?) 
x 
Ex. 4. ee 
w V(l = 2) 


Here we have to find the differential coefficient of 4/(1 — «*) with 
regard to x; we may arrange the work thus 


d 
feet) 1 — x2 1 1 
dy a/( x") ee WA x") x (1 — ¢7)? — ri(1 — x*)2( — Qa) 
dex ~ 1-2 i rk 


82 INFINITESIMAL CALCULUS CHAP. VI 


* 


EXERCISES VI(A). Inverse Functions. 


1. Show that the following pairs of functions f(x), 9(x) are such that 
Flo(x)] = a, o[f(x)] = a. 


; 1 2 1 
iL fa}= 225 o@) = 
» 5) 
ii, fx) = 9(e) = ——. 
n ive 
Lit u(e)e— er Sai (2) exe en (ora) 


Draw the graphs of f(x) and o(@). 

2. Prove that y = 1 — 2 cuts the axis of x at an angle equal to the 
angle at which x = 1— y® cuts Oy. 

3. Verify that (i) the gradient of y(~* + 1) = 6 at (1, 3) is the reci- 
procal of the gradient of y*% = 6 — x at (3,1), (ii) the gradient of 
y(« — 1) = 3x at (2, 6) is equal to the reciprocal of the gradient of 
(v — 3)y =x ‘at (6, 2), and (iii) the. gradient of y(2a —(])s=saeac 
(2 + 4/2, 2) is the reciprocal of the gradient of y = w + ~/(a — 2) 
at (2, 2 + 4/2). 

4, If the parabolas 

y = ax* + 2bu + ¢ x= ay" + 2by +c 
touch, show that (2b — 1)? = 4ac. 
5. Show that if the graph of f(x), traced upon transparent paper, is 


viewed from the other side of the paper, and the axes of w and y inter- 
changed, the resulting diagram is the graph of the inverse of f(z). 


EXERCISES VI(s). Differentiation. 


Differentiate the following functions 


1. (2a + 1) 2. (1 — 2x)? 3. (2 — 3x8 
4, 4/(2 — 32)8 5. (a + 1/a)? 6. 4/(% — 1/x) 
(1 + 2a)8 (1 — 3a)? joes f. 

i (1 + 3a)? a (1 — 2x8 ‘ VG + =) 

1 + 2? x x 
10. (3 u =) it) ee 12, TICE 


13. (1 + #31 — 2)? 9-14. (1 — x)®(Qa + 3)8 15. (w + a)P(a + by)! 


16, V(a* + 2) 17, Vie = 2”) 18. (ase) 


CHAP. VI EXERCISES 83 


19. (¢ zs =) jy alla nati Sy Yeti iano ral 
a-— 2 a/ (a + 2?) a/ (a* + a?) 
22. (1 + x)" + (1 — a)" Ps SA LI i ih el a A 8 gee nih 


~— 


24. J(at+ a2) +r4/(a- 2 
J(a+ x2) -+/(a - 2 


25. (a + 2bx + cau)” 


— 


26. (4a — 7)(3u + 7)° 27. (Sat + da + 3)4/(a® — 1)/28 

28. (1 + 2%)4/(1 — 2?)/x 29. (10 — 6x + 3a2)(5 + Qa)? 

30. (2e2 + 3)(3 — 5a®)? /x? 31. sintx 

32. sin? 2x 33. (sin hx)? 34. cos 2x 

35. +/cos 2x 36. sec? 2a 37. secta — tan*x 

38. sin’~ — cos*x 39. 4cos’z — 3cosx 40. tan33x 

41. 3tana + tan3x 42. seckx — 3seca 43. cos*x/a? 

44, x* cosec?x 45. tan?2(~ + «) 46. au 
4cos*x — 3 

47. (aes) 4g, £cose — 3 49. Ja — }sin 2x 

1 + sin x sin x 
50. 1 cos’ — cos x 51. gu + tsin 2x + 51, sin 4x 
52. « + cotxz 53. tanx — x 54. 1 tan’x — tanx + @ 


55. Differentiate x? with respect to 2° and wx? with respect to x’. 
56. Differentiate (ax + b)/(cw + d) with respect to (a’x + b’)/(c’a + d’). 
57. Differentiate 1 + x? with respect to 1 — 2?. 


58. Differentiate sin « with respect to cosw and sin 3x with respect 
to cos 22. 


59. Differentiate sin?x% with respect to cos®x. 


60. Differentiate tan x with respect to secx and secx with respect 
to tan x. 


(Cie heMe ime WIE 
TANGENT AND NORMAL 


68. Equations of the tangent and normal of y =f(x). 


It has been proved in Art. 33 that the gradient at any point on 
the graph of f(x) is equal to the differential coefficient of the function 
at the point considered. We may define the tangent at P as the 
straight line through P which has the same gradient as the curve at 
this point. Let P be the point (x, y,), and let the equation of the 


tangent be Y= 00-+.0 ee eee 
Then the gradient of (1) is equal to f’(x,) ; therefore 

a= f(x.) -. es 
Again, since (a, y,) lies upon the tangent, 


Y= 00 +10, ee 
Subtracting (3) from (1) 
Y — Yy = au — 2%) 
and the value of a from (2) gives 
yY — yy = f'(%)(% — x) 
as the equation of the tangent at (x, y,). 


Again, the normal is a straight line through P perpendicular to 
the tangent at P. The equation of the normal at P is therefore 


y-—Y, = — & - x) /f'(@) 
69. Illustrative examples. 
Ex. 1. To find the tangent and normal at (x, y,) of the curve y = x?/4a. 
Now f(x) = x*/4a ita ey Pie 
The equation of the tangent is 


Y — Y, = &,(% — x,)/2a 
2ay — 2ay, = x, — 2," 


LX, — 4ay, 


that is, 2a(y + Y,) = wa, 


CHAP. VII TANGENT, NORMAL 85 


The equation of the normal is 
Y¥ — Yy = — 2a(u — %)/ay 
which may be modified by writing y, = «,?/4a. 


Ex. 2. To find the tangent and normal of y = x® + 32? at the point 
whose abscissa is 1. 


Nowerf(ear 4 aa f'(%) = 32° + 6x 
Wetiuiuce resent Ljs.4 77(1) = 9 
The equation of the tangent is 
y —4 = ex —- 1) 
y= 9r — 5 

The equation of the normal is 

y-4= -3e-1) 

z+ Sy = 37. 


In an example such as this, which admits 
of representation on squared paper, the 
-student should draw the graphs of the 
function and the tangent on a largish scale, 
and convince himself so far as the test may allow, that the line touches 
the curve ; see Fig. 34. 


Fic. 34. 


70. The algebraic equation of a curve. 

It is often difficult, and sometimes impossible, to write the relation 
between the coordinates of a point on a curve in the form y = f(a), 
where f(x) is a simple function. Even the circle whose equation, 
a2 + y® — a® =0, may be written y = + 4/(a@? — x?) is dealt with 
more conveniently in the first form, and this is certainly true of the 
general equation of the conic 

ax? + Zhey + by? + 2gx + 2fy +c =0 
Generally, we shall write the equation of a curve in the notation 


where, in Section 1 of this book, we mean by f(x, y) a sum of a finite 
number of products of powers of x and y ; thus 


fica ye— Aaey +) Baeytre... Hany” 
Now, if we write Aer) 
we have 
dz ra d Man a P99 
iP = G, (Ax y”) + a, (Bury) “Pa ase 


d 
= Ama!" Aum Fyn + BpxP-tyf + Baur Ty! shi .ss 


86 INFINITESIMAL CALCULUS CHAP. VII 


Here y”, y%, ..., being powers of a function of x, may be differentiated 
by the rule explained in Art. 66, that is, 


d dy d dy 
PG NU es m—1 7% aed yy =k pacers 
da” ARR da” pth da 
It follows that 
dz / a : ( ah 
= m—1y,n—1 pel —1,,q-1 es 
oe Agm-ly ( my + nes) + Bar-lyi-l py + qua) + 
Again, since for all variations of x and y which satisfy 
f(x, y) = 0 
we have z = 0, we must also have dz/dx = 0; it follows that 
ig) + Bory (py + ane) 
m—1,,;n—1 ae: —1,,9-1 pad = 
Agel (my + nt a) + BaP-tyI-t| py + qua) + +. 0 


an equation which determines the gradient of the curve at (2, y). 


71. Illustrative examples. 
Ex. 1. To find the tangent of x? + y? = a* at (a, y,). 
Now, on differentiation we have 


Sat ty? —a*)=0 

that is, Den Oe ag 
dx 
Therefore dy Bote 
dx 7] 

The equation of the tangent is 
Y— Yy = — U(X — &)/Yyy 
that is, Lt, + YY, = Ae he OF = a2 


Ex, 2. To find the tangent of y* = 4ax at (x,, Y,). 
By differentiating y? = 4axv, we obtain 


The gradient at (x,, y,) is 2a/y,, and the equation of the tangent is 
Y — Yy = 2a(% — 2)/y, 
YY, — 4ax, = 2ax — 2a, 
YY, = 2a(v + 2) 
Ex. 3. To find the tangent of 2xy = c* at (x. y,). 
Differentiating 2xy = c?, we have 


I 
° 


es (xy) 


CHAP. VII SUBTANGENT, SUBNORMAL 87 


The gradient at (#,, y,) is — y,/x,, and the equation of the tangent is 
YY = — Wl — w)/ary 
ty, + Ye, = wy, = c 
Ex. 4. To find the tangent of the curve 2(a? + y?)? = 25(a — y)® at (3, 1). 
Differentiating, 


4(a? + 42) (a2 + y2) = T5(@ — 24 (@ — y) 
dx dx 


8(a? + P)(2 + y 2) = 75(x — ye(a ~ ) 


This gives the gradient at (x,y). Substituting x = 3, y = 1, the 
gradient m of the tangent is found to be 3/19. The equation of the 
tangent is epi tes BC 8) 19 

3x —- 19y + 10 = 0 


72. Subtangent and subnormal. 

We take P a point on acurve, VP its ordinate, 7'P its tangent, and 
PG@ its normal. Then 7J'N is called the subtangent and NG the 
subnormal. In the diagram a standard portion of a curve is drawn, 
in which 2, f(x), f’(x) are all positive. 


N 
HiGuso. 


We denote the angle N7'P by t, so that tan ) = f’(a), x being the 
abscissa, y = f(x). Then 


d 
TN =NPcotl =ycoty = f(a) /f'(x) = |< 


NG =ytan NPG = ytanyd = f(x) . f(a) = yw 


ty 
73. Parametric representation of a curve. 

The equation of a curve in Cartesian coordinates implies a corre- 
spondence between two sets of numbers, one of which consists of 


values of the abscissae of points on the curve and the other of their 
ordinates. But there are other geometrical magnitudes associated 


88 INFINITESIMAL CALCULUS CHAP. VII 


with each point ; thus, every point has a radius vector (r) and an 
angular coordinate (0), also the length of the arc (s) measured from - 
a fixed point of the curve up to the point considered, and the 
gradient (tan )) of the curve at that point may be regarded as 
coordinates, that is, as variables determined by the position of P. 
Concentrating our attention on the radius vector, there is a functional 
relation between x and 7, say x = F,(r), and, similarly, y = F,(r). 
The curve is completely represented by either of these equations, 


eg = I(r) y = f(r) 


It may also be represented by them both; for if we eliminate r 
between the equations, we must get the relation f(x, y) = 0, which 
we call the equation of the curve. The representation in terms of r 
of the two coordinates is not the only representation of the kind 
which is possible; by selecting 0, s, J), or some other variable which 
has a value at each point, we obtain other pairs of equations by which 
the curve may be represented, in each case the third variable is 
called a parameter. 

Perhaps the most general method of representation is obtained 
by thinking of a particle which describes the curve once, and once 
only ; we could then tabulate three sets of numbers, (1) the time (é), 
(2) the abscissa (x), and (3) the ordinate (y) of the point occupied 
by the particle at time ¢. The pairing of these would give 


a = f,(t) y = felt) 


The auxiliary functions vary, of course, according to the law of 
motion of the particle; they are, however, connected, for on 
eliminating ¢, we must obtain a functional relation constituting the 
equation of the curve, and this is independent of the rate of its 
description. 


74, Value of dy/dx, when x and y are functions of a parameter. 
Let ¢ = 7,0) 7 = f.0: 
Then x + 6x = f,(t+0t) y + oy = fot + dt) 


by _ fot + 6) - fo) _ falt + 08) - fo) | fid + 6) - fi 
dx f,(t + dt) — fi) ot ot 


If dt + 0, then dz — 0 and dy — 0, and we deduce 
dy _ fe _ dy [dx 


ey POY Be pak 


As a first illustration we take y? = 4ax, and write 


2 = ol y = 2at 


The curve is obtained by giving ¢a range of values from — © to o. 


CHAP. VII PARAMETRIC EQUATIONS 89 


Every quantity associated with the curve can be expressed in terms 


of ¢t, thus r = aty/(4 + &) tan@ = 2/t 
dy _ 4g = 
also tea at Malar 2a/2at = I/t 


Here ¢ may be called the cogradient, its value being cot v. 


A second important use of a parameter is in connection with the 
circle in which we take 


@ = @ cos 0 y =asin@ 
in this case 


dy _ Ata 
dz  d@/ d0 


which is geometrically evident. 


= acos0@/(- asin9) = -cot 0 


For a third illustration we have 
ay? = a7(a — 2) we ey Xx 
here we write x = a(1 — 7#), y = at(l — #?). 


The range of ¢ is [— 0, wo]; the range 
[- 0, —1] corresponds to the part of the 
curve in the second quadrant, since x is negative and y is positive ; 
{ - 1, 0] gives the fourth quadrant part, while |0, 1] gives the first 
quadrant, and when ¢ is in [1, © | we get the part of the curve in 
the third quadrant. 


Fic. 36. 


75. Equations of tangent and normal to the curve x=f,(t), y=f,(t). 


dy as fo © 
Now iG = AO 

It follows that the equations of the tangent and normal are 
respectively fly - AO] = Ole - f.0) 


and feOly - fo] = - fvOlz - fi] 


EXERCISES VII 


1. Write down the equations of the tangents of the following curves 
at the points mentioned 


Pe yon oe at (3,73) li, cy — Gat (— 2, — 3) 

iii, y = — 4/(5 — 4x7) at(1, -1) iv. y = +/(4a? + 5) at(1, 3) 

v. 13y = 23 — 3/x at (3, 2) vi. y = (% + 1)/(2u@ — 1)at(1, 2) 
vii. 6y = x® — wat (3, 4) viii. y = sin x at (1m, 4) 

Ixy = tan-2 at. (is, b) x. Y= sin a/e at (7; 0) 


2. Prove that 2a + y = 2aisa tangent to zy? = a®(x — a), and show 
that it cuts the curve again at (a, 0). 


90 INFINITESIMAL CALCULUS CHAP. VII 


3. Show that the tangent of 27ay? = 423 at (3a, — 2a) is normal to 
y* = 4a(x + 2a) at (-a, 2a). 


4. Find the equation of the tangent at (a, b) to 
(x/a)” + (y/b)” = 2 


Determine also the points other than points on the axes at which 
the normals go through the origin, distinguishing the cases of n even 
and odd. F 


5. Find the equations of the tangents and normals of 
(x? + a®)\(y — a) = 2a2x 
at the points (0, a) and (a, 2a). 


6. Prove that the tangent ‘to 2° + 2ay? — y? = 17 at (2, 3) cuts 
Oy at an angle of about 5° 42’. 


7. Show that 36x + 45y = ld4a cuts 2° + y® = az at right angles 
at (fa, 2a). 
8. Find the abscissae of the points of contact of the tangents of 
y = 8a® — 4472 + 78a — 45 
which are parallel to y + 2% = 0. 


9, Show that x27 + y? = 144 cuts y* = 10x at an angle equal to 
about 71° 1’. 


10, [he tangent at P fo y™ = kao mects OF at 4 rove tia 
mtan«?P =n tan TOP 
11. The tangent at P to y? = kz? meets the curve again at Q. Prove 
that tan OP =s02 tan VOxr 


12. The tangent at P. to. aty™ ='b"2" meets the axes in wi sandes 
Prove that the ratio of PT’: Pt is constant. 


13. Show that the length of the perpendicular from O upon the 


tangent at P (a, y) is dy Ny dy? aah 
(v a) (as) 


Prove that in the rectangular hyperbola this perpendicular varies 
inversely as OP. 


14. Give the Cartesian equation of the curve given by 
= acost y = asin 
Show that the tangent at ¢ is 
xsint + ycost = asintcost 
15. Prove that xcost — ysint = acos 2¢ is a normal to the curve 
given by x = acos*t y = asin*t 


Show that the locus of the feet of the perpendiculars from O upon 
these normals is r = acos 20. 


CHAP. VII EXERCISES 91 


16. Prove that 2a — 3ty + at® = 0 touches y? = az, 


17. Show that « = acos 2t, y = asin¢ gives a portion of a parabola, 
and that x + 4ysin¢ = a(2 — cos 2¢) is a tangent to this given portion. 


18. Find the equation of the tangent at ¢ to the curve given by 
% = at/(1 + 2%) y = all + ¢*) 
What are the tangents att = Oandt = 0? 
19, A curve is given by 
x = a(2cosé + cos 2t) y =a(2sinét — sin 2t) 
Show that the equation of the normal at ¢ is 


x cos $t — ysin $¢ = 3acos 3t 


Che beWe tee AW Ihe 
SECOND DIFFERENTIAL COEFFICIENT 


76. Definitions. 
The differential coefficient of f(x) has been written in two forms 


fe) Sf) 


When considered as a function of x, f’(x) is called the derived function. 
It is our purpose in this chapter to study the differential coefficient 
of the derived function. The totality of the values of these differen- 
tial coefficients is called the second derived function of f(x). The 
differential coefficient of the derived function is termed the second 
differential coefficient of f(x). The notations appropriate to these 
new conceptions are based upon the notations already used for the 
derived function and the differential coefficient. The second derived 
function may be written in either of the expressions 


d 
f'@) =f @) 
There are advantages in the second expression, as by its form it sepa- 
rates into two stages the difficulties which beset the generation of 
f(x) from f(x). For in any attempt to define f’(x) from f(x) we 


have to meet the difficulties which surround the definition of f(x) 
as well as those which occur in pee 


oy (x) 
By presupposing the existence of f’(x) the difficulties are encoun- 
tered singly. To define the second derived function, we must 


postulate (i) that f’(~) exists in an open range, situated within the 
range of definition of f(x), and (ii) that 


fi +h) -f(@) 
h 


approaches uniquely a finite limit as h +0; if these conditions are 
complied with, we may write this limiting value as 


£ f(a) or fe) 


CHAP. VIII SYMBOLIC NOTATION 93 


It should be noted that the second condition implies the continuity 
of f’(x). 

The corresponding notation for the second differential coefficient 
is built up on the analogy of the forms by which we represented the 
first differential coefficient, namely, 

dy a , 
ie dni) = f'(x) 


Thus the second differential coefficient is 


d di hae ' 
Se Eagle -f@ 
the first expression is written more succinctly as 
ay 
dsc” 


an expression which is read as ‘d two y by dx squared.’ 


77. Symbol D for differentiation. 


We may now introduce a further simplification in our symbols, 
by writing D for the operation of differentiation. Thus, in the case 
of a single differentiation, we write 


d d 
Df) = f@) Dy = 5" = 5 fea) 


and in the case of a second differentiation 

Gale 1h. Ie) EM ae eS. 
dy addy dy 
dx dudx dz? 

Further applications of the use of D are given in the following 
chapter; but at the present stage it may be regarded as a short 
way of writing an operation. Sometimes a letter is suffixed to 
indicate the variable with respect to which the differentiation is 
effected ; thus D,y signifies the differential coefficient of y with 
regard to x. The result of Art. 62 would be written 

D,y . Dz = 1 
and that of Art. 74 would be 
Dig Diy Dix 


It is permissible, with requisite safeguards, to extend the process 
of differentiation and to obtain differential coefficients of higher 
order ; thus, we have 


D’y = D 


71 TeeeO y - dy dl 
3 — —_ = =~ 4 = — n = — 
peciiea Die a8? ee dx’ ** i da” 


94. INFINITESIMAL CALCULUS CHAP. VIII 


It is clear that two of the laws of positive integral indices hold for 
powers of D. Thus D"Dry = Dmtny 


and (D™)"y = Dy 


78. Geometrical meaning of the sign of the second differential co- 
efficient. 

The argument of Art. 42 may be repeated by the substitution 
of f’(x) for f(x). Thus, if f’(~) increases with x, Df’(x) is positive ; 
again, if f’(x) decreases with x, Df’(x) is negative; also f"(~) = 0 
gives the values of x for which f’(x) is stationary. 


We now consider the geometrical properties of the graph of f(a), 
which are indicated by the sign of f’(x). These could be deduced 
from diagrams such as Fig. 13, p. 52, where typical graphs of f(x) 
and f’(x) illustrate this problem. For the interest of the question 
we follow another course, and consider the rotation of the tangent 
as P, its point of contact, describes the curve. 


Let us draw lines Op, Oq parallel to the tangents at P, Q in 
Figs. 37a, 376. Although in both curves there is an up grade, 


Fic. 37a. Fic. 370. 


the tangent is in the two diagrams rotating in contrary directions. 
In the left-hand diagram the tangent, and therefore Op, is rotating 
counter-clockwise, while in the second the direction is clockwise. 
The obvious geometrical distinction between the curves in the two 
diagrams is that the left-hand curve is concave to an observer 
situated at a great distance above Oz, and that the right-hand curve 
is convex. Thus, concavity at P in Fig. 37a corresponds to the fact 
that at P the tangent is rotating counter-clockwise, as the abscissa 
of P increases. 

Now, in a curve such as the graph of the quadratic function, the 
tangent is always rotating in the same direction : the curve, therefore, 
is concave or convex throughout ; but other curves which are the 
graphs of single-valued functions may change from concave to convex 
or from convex to concave, as x increases ; an example of such a 
curve is the well-known sine-curve. Let us see what happens at 
the junction of convex and concave portions. 

Let us take a curve APB, in which the arc AP is concave upwards 


CHAP. VIII POINT OF INFLEXION 95 


and PB is convex, and study the rotation of the tangent as its 
point of contact moves from A to B. The tangent revolves } from 
Oa to Op and then ) from Op to Ob. The 
gradient of the tangent at P is a maximum. 
The tangent at P differs from an ordinary 
tangent, inasmuch as in the neighbourhood 
of P the arc AP lies above the tangent 
and the arc PB lies below it; the curve 
indeed at P crosses its tangent—such a 
point is called a point of inflexion. There 
is also a point of inflexion at which the 
gradient is a minimum; such a _ point 
separates a convex portion on the left from a concave portion on 
the right. The student should draw a figure to illustrate this case. 

The conic sections have no points of inflexion, but such points 
are familiar to the skater as points on the ice-curves described by 
his skate at which he changes his edge ; another example is afforded 
by the familiar figure 8. 


Fic. 38. 


79. Relation between the graphs of f(x), f(x) and f’(x). 

The relation between the graphs of f(x) and f’(x) has been discussed 
in Art. 46. It remains to discuss the relation between those of 
f(x) and f’(x) and to see what properties of the graph of f(x) are 
revealed by properties of the graph of f"(z). 

Now, when f"(x) is positive, f’(x) is increasing and the graph of 
f(x) is concave in the sense explained above, while if f’(x) is negative, 


Fie. 39. 


the graph of f(x) is convex. Also at a node of the graph of f’(z) 
we have a point of inflexion of the graph of f(x). Again, at an 
ascending node of the graph of f"(x), the graph of f(x) changes from 
convex to concave, and at a descending node, the graph of f(x) 
changes from concave to convex. 

Another combination is of interest ; the graph of f’(~) may touch 
Ox but not cross it. In this case the graph of f”(x) passes through 


96 INFINITESIMAL CALCULUS cmap. vit 


the point of contact of the graph of f’(x) and the graph of f(x) has 
a point of inflexion at which the tangent is parallel to Ox. 


80. Criteria for maxima and minima. 

It is easy now to give a second set of criteria for stationary values 
which are maxima or minima. We see that 

(i) if f’(a) = 0 and f"(a) is positive, then f(a) is a minimum ; 

(ii) if f(a) = 0 and f(a) is negative, then f(a) is a maximum. 

But if f’(a) = 0 we cannot discriminate the stationary values 
without reference to higher differential coefficients. 


81. Illustrations of f’(x). 
First, we take the quadratic function 
f(x) = ax* + 2bx% +c 
f(x) = 2(ax + b) 
et "24 
The graph has no point of inflexion, it is concave to an observer 
above Oz, if a is positive ; convex, if a is negative. 

Secondly, we take the cubic function 

f(a) = ax? + 362? + 3c~ +d 

f(x) = 3(ax* + 2bx + 0) 

f'(x) = 6(ax + b) 
The graph of the cubic function has a single point of inflexion 
which lies upon x = — b/a: this line divides the graph into two 
parts, one of which is concave and the other is convex. Further, 
if b? = ac, the inflexional tangent is parallel to Ox, because in this 
case f’( — b/a) =0. 

Just as the quadratic function was reduced to the form ax’® + D/a 
by writing w’ = x + b/a, that is, by taking the origin on the axis of the 
parabolic graph; so, by choosing x’ = x + b/a, the cubic function 
may be reduced to a form in which the second term is absent. In 


the reduced form of the cubic the new axis of y passes through the 
point of inflexion. 


82. A parabolic approximation to y=f(x). 
In Art. 68 we determined a linear function 
ax +b 
so that its graph passed through a point of 
y = f(z) 
and had the same differential coefficient as that of f(z) at the given 
value. So we can now determine a quadratic function 


ax + 2b4 +c 


CHAP. VIII CIRCLE OF CURVATURE 97 


such that its graph passes through the point (a, f(v,) ) and has the 
same first and second differential coefficients there as the given curve. 


It follows that f(@) = av + 26x, +c 
f (@) = 2(ax, + D) 


and these three equations provide means of determining a, b, ¢ in 
terms of x, f(x,), f’(~%,) and f’(v,). Geometrically, this means the 
determination at x = 2, of the parabola of closest contact which has 
its axis parallel to Oy. 


83. The circle of curvature. 


It is more convenient to compare the curve y = f(x) at each 
point with a circle than with a parabola. The comparison still in- 
volves the determination of three constants, namely, the radius and 
the coordinates of the centre; these are settled in the same way as in 
Art. 82 by making the circle pass through the point considered 
and have values of Dy, D?y equal to those obtained from the equation 
of the curve at the point. 

Let the equation of the circle be 


(aati) te (fee hye aa cA te ee CL) 


where (h, k) is the centre and c is the radius ; then by differentiation 
we have 


@ -h +y-Y =0 SO (2) 
and by a second differentiation , 
d dy dy 
Bits Ua) i tat k) 3 0 
; sey ay 
that is, 1+(2) +@-mho4-0 .... 8 


Now, from equations (1), (2) and (3) we find h, k, c in terms of 
x, y, Dy, D*y ; thus, for h, k we have 


= Ny ee al - -| (2 
o-h=l[1 + (3) | Ue eed ord Nar 


and substituting in (1), 
dy}? | anil ayy 


dy\? ee 
hence c= + E + ean Ta 


To make ¢ positive, we select the upper sign if D*y is positive, 
and the lower sign if D®y is negative. The circle whose centre 
(h, k) and radius c are thus determined is called the circle of curvature 
at (x, y). 


C.c, G 


| 


98 INFINITESIMAL CALCULUS CHAP. VIII 


84. Curvature. 


A brief statement, which will be supplemented in Section II, is 
made of the meaning of curvature of a curve at (a, y). 

In a circle the curvature (or bending) at every point is the same ; 
this is an inference from symmetry. Again, in a large circle the 
curvature at each point is small; in a small circle it is large. We 
say that the reciprocal of the radius of a circle is a measure of its 
curvature. Also the reciprocal of the radius of the circle of curvature 
at a point (x, y) is a measure of the curvature of the curve at this 
point. 


85. Formulae for curvature. 
Although the radius of a circle is a positive quantity, it is con- 
venient to give the curvature the sign of D?y and take for its value 
ay 
] dx” 


e dy\2 2 
eae 
| * dex 
With this convention we see that positive curvature implies that the 
curve is concave upwards, and negative curvature implies that it is 


convex. We may notice also that at a point of inflexion the curva- 
ture is zero. 


The formula for curvature assumes a very simple form when the 
tangent at P is parallel to Oz ; in this case the curvature is 
Lage 
py das 
Further, in the case in which the gradient is small throughout, as, 


for instance, in a beam suspended by its middle point, the curvature 
may be taken as equal to 1 dy 


0. dx 
since in the denominator of the expression for curvature the gradient 
is small and only its square appears. 


86. Illustrative examples. 
Ex. 1. To find the radius of curvature at any point of the parabola 
y = «/4a. 
We have Dy = «/2a, D?y = 1/2a and 
1 + (Dy)? = 1 + a/4a?=1 + y/a 
hence the radius of curvature = 2(@ + y)® /a®. 


Since in the parabola a + y = SP, the square of the radius of curva- 
ture varies as the cube of the focal distance. 


CHAP. VIII , EXAMPLES | 99 


The diagram, Fig. 40, represents circles of curvature, the abscissae 
of whose points of contact are 0, a, 2a, 3a,... , the centres of the circles of 
curvature, or centres of curvature, being at Co, C,, Cy, ...; the parabola, 
though not drawn, is suggested by these circles of curvature. 


C, 
C \ 
S \ lf P, 
P 4) 


oJ 


Fic. 40. 


Ex. 2. To find the radius of curvature at any point of the ellipse 
aja? + a*/b* = 1 


By differentiating ba? + aty? = ab? 
dy 
e have 2 oe dee 
Ww b*x + ay rE 0 
: dy 2 d?y 
Iso 2 2 op beast A 
als b + a2 (3) + ay 7s 0 
2 2 4 
whence oe ie Pee ee OS 
dx a*y AMEN Vahey: 


The radius of curvature is equal to 
(aty? + b4x2)? /atb! = CD3/ab 
where CD is the semi-diameter parallel to the tangent at (a, y). 
Ex. 3. To find the radius of curvature at the point (1, 2) nf the curve 


whose equation is 
: y* = 2 + 32% 


100 INFINITESIMAL CALCULUS CHAP. VIII 


By differentiation we have 


2y = = 3x2 
Sh ae ou" + 62 
dy\2 d*y 
d = i fe 
an 2) + ay 7a 6x + 6 


Writing « = 1, y = 2, we find 
dy 9 d*y 15 


dx 4 da® 32 
whence c = 977/30 = 31-85 


87. The orders of magnitudes. 

There is no absolute standard by which a particular quantity can 
be said to be small. Thus, in measuring the radius of the earth a 
mile is small, while in measuring a man we take account of the 
quarter of aninch. In the infinitesimal calculus it is with the ratio 
of dy : 6x, where x and y are functionally related, that we are so often 
concerned ; these small quantities of the calculus have, however, a 
property which renders them particularly difficult to grasp; it is 
their liability to variation. For when we are discussing the ratio 
of dy: 6x, we must think of the ratio of the terms of a sequence 
of dy’s to the corresponding terms of a sequence of dws. Let us 
suppose that the limiting value of dy: dx is m, and let us consider 


the sequence 6,7, 6,7, 6,v, ..,, and the consequent sequence of 
the increments of y, dy, doy, Osy, ...; then we can write 

Oy Oey OsY 

Piola all ck Gk Stas Sv.) ee 


where the e’s are small compared to m, and by the nature of m, the 
e’s form a sequence whose limit is zero. In this case we say that 
the small increments dz, dy are of the same order. 

Two cases have to be further considered, namely, when m = 0, 
and when 1/m =0. Under either of these conditions dx and dy are 
of different orders. 

If m = 0, then, we may find that 


oy 
(6x)? =n + € 

where n = lim dy/(6x)? and lime = 0, 6x +0; in this case, the 
order of dy is double that of dx. Generally, if 


(dy)? 
xyt = k+e 


where k is finite and not zero and e +0, when dx -> 0, we. say that 
dx is of the pth order of small quantities and dy of the gth order. 
By this we imply that if the typical small quantity is 10~, dy is of 


CHAP. VIII ORDER OF MAGNITUDE 101 


the order 10-7, and 6a of the order 10~?, whence it follows that 
(dy)? and (6x)2 are of the same order, namely, 10-72, 


As a geometrical illustration let us take a curve which passes through 
O and represents a functional relation between # and y. Then (6a, dy) 
is a point near O. If 8 
ua 
> =mte 
the tangent at O is inclined to Ox at an angle tan-!m. But if m = 0, 
Ox is the tangent at O, while if 1/m = 0, Oy is the tangent at O; in 
these cases dx, dy are of different orders, and their respective orders 
depend upon the equation of the curve. 
If the equation of the curve is (1) y? = aa, then (dy)? = ada, and 
dx being of the second order, dy is of the first order (Fig. 41a), while if 
(2) it is v? = ay, da is of the first order and dy of the second (Fig. 41b) ; 


Fic. 41. 


again, if (3) it is y3 = a’w, then, dy being of the first order, da is of the 
third order (Fig. 4lc), and if (4) 73 = a®y, dx being of the first order, 
dy is of the third order. The diagrams give points whose abscissae 
are 0-1, 0-2, ..., a being taken unity. 

Further information on the geometrical form of a curve at a point is 
given in Chapter XVII. 


We may also illustrate the important fact that there are different 
orders of large magnitudes by taking the equation of the hyperbola 
referred to a vertex as origin and by supposing that the lengths of the 
axes of the hyperbola become infinitely large, while its eccentricity 
approaches unity as a limit, deducing from it the equation of a parabola. 

The equation of the hyperbola referred to an end of the transverse 


axis as origin is (© + a)2/a® — y2/b? = 1 
that is, y® = 2xb?/a + x*b*/a? 
and e” = (a7 + b?)/a? = 1 + 6 /a* 


Now, if a—o, b-~o, and e—1, we must have b?/a* — 0, so that 
b/a —0; this requires that a, b are of different orders. If we take 


102 INFINITESIMAL CALCULUS CHAP. VIII 


that b?/a — p, a finite magnitude, we have as the equation of the curve 
approached y* = 2xp 

The conditions imposed mean that while b is of the first order of large 
quantities, a is of the second order. 


88. Velocity and acceleration. 

If 2 is the coordinate which determines the position upon Ox at 
time ¢ of a particle describing this axis, and if x + dx gives its 
position at time ¢ + dt, then dx/dt is its average velocity in the 
interval dt. Again, the velocity at time ¢ is lim 6a/dt, dt > 0. We 
write this limit dx 

dt 
The notation of the differential coefficient is naturally adapted to 
the measurement of velocity. 


Again, if v + dvis the velocity at time# + dt, the average accelera- 
tion in the interval 6¢ is 6v/dt, and we have 


dv 
| anf 
where f is the acceleration at time ¢. 
Also we have, in the notation of Art. 76, 


dv ddx dx 


Dash FE mIHGTy mo Br. 

Valuable illustrations of the calculus are derived by studying the 
graphs of x and v considered as functions of ¢. The first is called 
the displacement-time graph, and the second is the velocity-time 
graph. They are connected with each other by the relations which 
unite the characteristics of a function and its derived function. 
When the velocity (v) vanishes, the particle is at a stationary point ; 
if v not only vanishes but changes sign, the particle is at a maximum 
or a minimum distance from O. Further, if the graph of f is drawn, 
we see that, at the stationary points of the v-graph, the acceleration 
of the particle vanishes, and that at the corresponding point of the 
a-graph there is a point of inflexion. 

Turning to the diagram on p. 95, the reader will see that if times 
indicated by the abscissae of B, C, ... be denoted by the corresponding 
small letters, the velocity vanishes at times b, c, d, e, f,..., the 
acceleration vanishes at times p, g, 7, d, 8,t,..... The point D, which 
is a stationary point and a point of inflexion, corresponds to a dead 
point of the motion, since there is no velocity and no acceleration. 


89. Leibniz’s theorem. 
If uw, v are functions of x, we have, from Art. 35, 
d ay dv du 


CHAP. VIII LEIBNIZ’S THEOREM 103 


Or, as it may be written, 
D(uv) = uv’ + u'v 


where dashes denote differentiation with respect to the variable 2. 
This theorem can be generalised, and we can find D"(wv). First 
we find D?(uv), thus 


D?(uv) 


Diu’) + D(w'r) 
uv” +uv +uv’ + uv 
= uy” + Quo’ + wv 


I 


Let us assume that 
D* uv) = w*—) + (n — 1)’ + (n — Igu%in-?) +... 
+ (n — 1)yul™y’ 4 uly 
where 7, is the number of combinations of different things taken 


r together. 
Now, on differentiating each side of this identity, we have * 


D”(uv) = yy) 4 w’ynr-) ae (n the 1), (u’yin-) ae uryn-2)) 
+ (nm — 1),(u"'m—2) + w’ol"-3)) 4 + ulD Dy’ + uly 


w™ + [1 + (n — 1),] wo) 
+ [(m -— 1), + (m — 1),| uo") + 2. + wld 


= uy” + nw") + nyu") +... + uly 


The result thus established by induction is named after Leibniz, 
one of the great thinkers who laid the foundations of the infinitesimal 


calculus. 


EXERCISES VIII (A) 


Second differential coefficients, points of inflexion 


d?y 
Gan ) ae ae 1) 

avwz+be+e dy 2%(a+b+4+¢) 

ni eat yh 
oe dy 2 4. 
sand hele GT (oo 2) dt (@e-ip” & — 2 
/ 

Liye ee bet e Py _ Aap* + bp +0) 2(ag? + bg + 0) 
Be Yy pj — — 


(x — p)(w@ —q) da (p—q)(x-pP (p-q)(«—q)> 


* The theorem used in reducing the coefficients is 
N,=(N—-1),_-1+(n—-1), 
a result which is directly proved by considering the combinations of the group of n 
different things as compared with those which are made first by excluding a particular 
thing and then by including it. 


104. INFINITESIMAL CALCULUS CHAP. VIII 


2 d*y 2-3 
Salt ay n/t) qa = (+ #*) 
d* 
6. If y = xsinx + cosa Fa3 = COS@ — wsins 
2 
7. Tee se oma Ty eas © 
COS @ dx? (1 — sinz)? 
d? . 
8. lf y = Asinz + Bcosx — -y 
t d*y 
9. If y? = sec 2x qa = Bh -y 
d*y ab 
10. If y2 = aa? +b i i tre 
Yy ax? + alee 
2 2 2 d*y | hee es 
Deals bores ee etl re ay ie 
4 dat wu, 3 uke a 


12. Show that y = 2723 — 54x02 + 36x — 8 
has a point of inflexion on Oz. 
13. Show that y = 24 — 622 + 5x 
has points of inflexion at (1, 0) and (— 1, — 10). 
14. Determine f(x), a cubic function of x, such that y = f(x) has a 


point of inflexion at (— 1, 16) and touches Oz at (1, 0). 
Ans. x? + 302% — 9x + 5. 


15. Determine f(x), a quartic function of x, such that y = f(x) passes 
through O and has points of inflexion at (1, 7) and (-— 1, — 17). 
Ans. 2* — 6x2 + 122. 


16. Show that the points of inflexion of 
y = 4sina — sin 2x 
are at points whose abscissae are nz, (2n + £ @)™ where n is integral. 


17. Show that the origin, (4/3a, hax/3) and (— 4/3a, — 4a,/3) are 


the points of inflexion of y(a? + a2) = ate 


18. Prove that the abscissae of De points of inflexion of y?(1 + a4) = 2? 
AT ome, wer b. 
19. Show that y = zy + y* + 2° has a point of inflexion at O. 


20. Show that y= (a — a*)/axz has a point of inflexion at one of the 
points at which it cuts Oz. 
‘21. Ife = cosé + tsin#, y = sint — écost, prove that 
dy d*y sect 


ad, aay ne, 
dx ae da? t 


22. lig= t—t)) y= vee prove that 


CHAP. VIII | EXERCISES 105 


23. If x = sin y, show that 


d?y dy 
en gee |p eerie 
Ce dx2  * dex : 
24. If a = siny/y, show that 
dy dy 
— rh Veet Ee —_ = 
(1 — @*) eats a 2 


25. By applying Leibniz’s theorem to the case in which u = a”, v = a”, 
deduce that the sum of the squares of the coefficients in the expansion 
of (1 + x)" is equal to (2n)!/(n!)?. 

26. Find the nth differential coefficient of x” log a. 

1 n(n —1) 1 n(n — 1)(n — 2) 
Ans. n\} 1 — = fF 8 
ns. nN Joga + 5 ry + 5 31 «| 

27. Show that 

i. D"xf(x) = af'™ (x) + nf'"-(z) 
li, Dx? f(x) = afar) + 2naf'"—"W(x) + n(n — 1)f'"-?(x) 
28. Prove that 
i, Dx cosa = xcosa% + 4nsinz 
ii, D*"+1 42 sin x = a cosa + (8n + 2)xsina — (16n?2 + 4n)cosz 

29. Prove that if 2 and y are connected either by the relation given 
in Question 23 or by that in 24, 

(1 ie 2 )yln+2) 2 (2n a 1) ayer) Pe ney) = 0 


EXERCISES VIII (8) 


Radius of curvature 


1. Construct the circles of curvature of y* = x at the points (0, 0), 
(1, 1), (4, 2), ... by calculating the radii of curvature at these points and 
determining the corresponding centres of curvature. 


2. Compare the radii of curvature of the curves 
y2 = a3 y= 2 
at the three points whose abscissae are 1, 4, ;4,. Show that at the 


origin the radius of curvature of the first curve is zero and of the 
second is 0. 


3. Draw the circle of curvature of the curve 
y=u2+i/zx 
at the point (1, 2), and show that p = 4. Prove that the parabola of 


closest contact at this point with axis parallel to Oy is y = a — 2x + 3. 
Draw a diagram showing this parabola in its relation to the curve. 


4, Sketch the curve y= re 


and find the circles of curvature at the points given by x = — l, 0, &, 


Ae: 


106 INFINITESIMAL CALCULUS CHAP. VIII 


5. Draw the parabolas 
y= —-24+a4+1 a= —-ytyt+l 
and show that they have the same circle of curvature at (1, 1), namely 
telah ERAS 
6. Prove that the parabola 
2y = (yy — Wy + Ys)(a/h)? — (yy — Ys)(x/h) + 2Ye 
passes through (— h, y,)(0, y.)(h, ys). Deduce that at (0, y5) 
Dy = — 34, — ¥3)/h = D®y = (yy — 242 + Ys)/h? 
7. The coordinates of three points P, Q, R on a curve are 
(0-2125, 0-8734) (0-2250, 0-9321) (0-2375, 0-9914) 
Find approximate values of Dy, D?y at Q, and deduce that the radius 
of the circle PQR is 29-2 nearly. 
8. Given three points P, Q, R on a curve 
(a, Y,) (a +h, Yo) (ath+k, ys) 
show that an approximate value of the second differential coefficient 
at P 1s 9 kyy — (h + kyo + hys 
hk(h + k) 
Given h=1, k = 2, y, = 0-312, y, = 0-451, y, = 0-675, show that 
the radius of the circle PQR is about 56-82. 


9. Show that the radius of curvature of 
y = asin (a/b) 
at x = $nb is equal to b?/a. 
10. Show that the radius of curvature of y = 4sina - sin 2% at 
x = 4n is 54/5/4. 
11. If the radius of curvature of the ellipse x?/a? + x?/b? = 1 at the 
point (a//2, b/4/2) is a, prove that the eccentricity of the ellipse is 


V/(3 — v5). 
12. Prove that the radius of curvature at (a, b) of (w/a)" + (y/b)” = 2 
is equal to (a2 + b2)? 
2(n — l)ab 
13. Show that a? + y? = 2, at + y* = 2 touch at (1, 1) and that their 
curvatures at this point are in the ratio of 1: 3. 
14. In the curve y? = av’ + 6b? show that 
_ = (y? + ax)? fab 
15. In the curve y2(2a -- 2) = 2 show that 
_ av/x(8a — 3x)! 
i nl 2a, ale 
and that the centre of curvature is at 
ax(5x — 12a) 8 ~) 
Claes x 


CHAP. VIII EXERCISES 107 


16. In the astroid w* + y> =ai prove that 
0 = Bat at yt 
17. Given a curve in which = cost + ¢tsint, y = sint — tcos#, 
show that p = 2. 


t t? 
18. Gi th te = ——s 
iven tha 4 Tanz Yy rey: 
dy 2t d*y 21 + #)8 


show that 


de l—-P de2 ~“(1—#- 


Verify from the value of p that the curve is a circle. 


EXERCISES VIII (C) 
Motion in a straight line 


1. Sketch the displacement-time and velocity-time graphs of the 
following motions in one dimension, mentioning discontinuities in the 
second set of graphs 


i, A ball falling to the ground. 
ii. A ball moving up and down a billiard table (e = 3). 
ili. A pendulum executing harmonic oscillations. 
iv. A mass dropped into a bucket of water, considering the cases in 
which the mass is lighter or heavier than water. 
v. A train going from one terminus to another and back, with a stop 
halfway. 
2. Find the velocity at time ¢ = 0 and the acceleration, given 
x = at? + 2bt +c 
3. Find the maximum velocity and the positions of instantaneous 
rest in the motion a = a(sin nt + cos nt) 


4. If the relation connecting displacement (wz) and time (¢) is 
2ut = ax* +-2be + ¢ 

show that the velocity varies inversely as the distance from the point 
x = — b/a and that the acceleration varies as the cube of the distance 
from this point. 

5. Given v2 = n*(a? — a), show that the acceleration is directed 
towards O and varies as the distance from O. 

6. If the relation between 2 and ¢ is 

2t 
e+e 
show that the velocity vanishes when ¢ =a and the acceleration 
when ¢ = av/3. 


Gln beet hee D6 


INVERSE DIFFERENTIATION 


90. General remarks. 


In an earlier chapter the relations between functions and their 
inverses were discussed. The simple functions, such as a + 2, aa, x?, 
depend upon a single mathematical operation ; their inverses x — 2, 
a/x, s/x also imply a single operation, which may be termed the 
inverse of the former. Thus, addition and subtraction are inverse ; 
so also are multiplication and division, and again, involution and 
evolution. The characteristic of inverse operations is that if they are 
successively applied to x, they reproduce x. Thus, if 2 is added to 
x, and then subtracted from the sum, the result is x; also, if xis 
multiplied by a and the product is divided by a, we get x; further, 
the square of the square root of x gives x. In single-valued opera- 
tions it is indifferent which of the operations is applied first ; thus, 
not only does (x — 2) + 2 =a, but also (vw + 2) -2 =u. 

The operations addition, multiplication and involution are called 
direct, while subtraction, division and evolution are inverse. The 
importance of these inverse operations may be realised by 
reflecting that negative numbers are due to the need of sub- 
tracting the larger of two positive numbers from the smaller, 
and that fractions are the outcome of our need to extend the realm 
of division. 

As inverse operations are not only more difficult, but more 
tentative, we may expect, in turning to the process which is the 
inverse of differentiation, to find new difficulties and also to require 
considerable additions to be made to the list of functions which 
have been studied hitherto. 


91. Definition of inverse differentiation. 

The problem of inverse differentiation is expressed thus: Given 
a function of x, say f(«#), it is required to find a function whose 
differential coefficient with regard to x is f(x); that is, given f(a), 
it is required to determine y, a function of 2, to satisfy the equation 


dy 
ia, f(z) 


CHAP. IX INVERSE FUNCTION 109 


As a first illustration of the indefinite character of the answer we 
take o(a) = 2%,and ask for functions which when differentiated give 2a. 
The answers are numerous, and may include 


x ew +1 1 ole Bee 
indeed, any function which differs from x? by a constant. The complete 
answer is represented by y= 2+ 


where C may be any constant or fixed number. 


92. Geometrical meaning of inverse differentiation. 


We have shown in Chap. VI the importance of studying the 
graph of the derived function f’(z) side by side with that of f(z). 
When we are given f(x) there is a single definite derived function, 
but when we start with f’(%) as given, there are many curves whose 
gradient at a point whose abscissa is 2 is represented by f’(x). For 
if one such curve is drawn, we can get any number of other curves by 
shifting this curve so that each point moves parallel to Oy through 
the same distance. 


Taking the illustration of Art. 91, 

d 

= (L) =) 22 

Ste) 
we have f(x) = a? as one answer. The 
gradient at P of the parabola y = 2 is 
equal to the ordinate NQ, but obviously 
if the parabola is shifted up or down, 
parallel to Oy, the tangents at P and P’ 
are parallel, and their gradients are both 
equal to NQ, that is 


f(@) = a7 + C 
is also an answer to the question proposed, where C is the distance 
parallel to Ox through which the parabola is shifted. 


Fic. 42. 


93. Notation of an inverse function. 


The result of certain mathematical operations performed upon x 
is denoted by f(x), and there is an operation which, performed upon 
f(x), reproduces x. This operation is denoted by 


f(a) 
Then, on substituting f(x) for x, we have 
| fUf@)] = x 


which is in accord with the analogy of an index law. 
The reader may compare the notation 
SISifia aaa sid-1 sine. = 
and should note that in the second equality a selection has to be 


made of the appropriate value of the incompletely defined function 
Sy nkete 


110 INFINITESIMAL CALCULUS CHAP. IX 
Again, if f@))> 2%, fra) = x?, we have 
fia) = fle) =)? = « 
ffl) = fA@?) = (a)? = a 
selecting properly the sign of the incompletely defined square-root 
function ; thus ff- 1) =f3d) = Leia 
Generally we have that 


ALAAM@] = & = FAF@)] 


94, Notation of inverse differentiation. 
We have used the notation 


Dftay = f(a 0 2 ee 
It is a ready deduction from this to say that 
f(@) = D-41f'G@) | 2 ace 
Again we have from (1), by applying D7! to each side, 
Dai Dia) Dr fay a) ae 
and by applying D to each side of (2), 
DD-f'(a) = Df@) =f@) ... . @ 


where the results in (8) and (4) are consistent with our general 
inverse notation. 

We use D-! as asymbol for inverse differentiation, just as we have 
used D for differentiation, but it does not completely define an 
operation. Thus we may write 


UD eh phat fa 
as a consequence of Da? = 24 
but we also have Dai2re on a 
as a consequence of Die" +1) = 22. 


This incompleteness of definition is allowed for by writing 
De oa Aye Gd 


We shall suppose that in effecting D-! the constant is chosen so that 
the result of performing first D and then D-! upon a function 
reproduces the function. Symbolically we shall take D-!D = 1 
as well as, DD7! =) 1; 

The operation of inverse differentiation is usually termed inétegra- 
tion, and the result of integrating a function is spoken of as its 
integral ; sometimes the function to be integrated is called the 
integrand. — 


CHAP. IX INTEGRAL OF A POLYNOMIAL lia 


95. Integral of x", provided n + — l. 


Now = yn 
n+l 
gyntl 
It follows that ‘Dyetig ae 
n+l 


A rule may be expressed thus: to integrate 2”, add unity to the 
index and divide by the increased index. : 


As particular examples we give 
a 1 
Drei ria DDFS ew) a es tee) eg ee 


<1 3B 
LOTR Wie = D-ly? = 27? D-1,° =-2 


3 
96. Integral of a sum of two functions. 
We have to show that 

Du + v) = D-*w + D-o 
Now we know that DD“(u4+v) =u+v 
and that D(D-1u + D-v) =u +v 
therefore DD-l(u + v) = D(D-1u + D-!v) 
It follows by operating upon both sides with D~! that 

D-7(u +:v) = D-4u + D-v 
The extension of the process to the sum of a finite number of integra- 


tions is easily made. 


97. Integral of the product of a constant and a function. 
We have to show that D-!Cu = CD-!u 


Now DD-'Cu = Cu 

and DOD Sui CDD +a = Cu 

Hence DD-1!Cu = DCD-!u 

It follows by operating upon both sides with D-! that 
DCU, = CD-ty 


98. Integration of a polynomial. 
Let f(x) = axv™ + ba® 14+... +he +k 
By Arts. 96, 97, 95 we have 
Define Dae out. the +k) 
= Dax” + Dbz" 1+... + D“he + Dk 
aD-14" + b6D-te"1+... + AD“ ax + kD-1x° 


a b h 
= —gerl + —-9h +. + ae + ke + OC 
n+l n 2 


112 INFINITESIMAL CALCULUS CHAP. IX 


99. Integration of a general polynomial. 3 
Let P(v~) = ax + bx? +... + kar 
where a, 8, y ...x have any values except - 1. Then 


-—1 — tL Gs _yatl b 
D-1P(z) z + 8 + 


k 
Bt 4+... + —— axti.+ C 
a+l 1 rane 


100. Integration of a function of a function of x. 


No general statement of the solution of this problem can be made, 
as it is only a small number of compound functions that can be 
integrated. A few illustrations are given of the integration of 
certain compound algebraic functions. 


dy 

Ex. 1. To solve ee = 

dx AAR ot 

We change the variable so as to reduce the integrand to a power of 


the new variable. Writing peal Aer 


Cyt 
ee 
But we must also change the independent variable from z to z on the 
left-hand side. This is effected by writing (see Art. 66) 


dy  dydx 
dz dx dz 
dy _ 3 4 
We now have = i 270 V) Se: 
dz 
Integrating with respect to z, 
pS — 228 + O = = 21 ay 


It may be noted that we have here a particular case of the equation 


Neon Fe 
pe hte tho) 


the solution of whichis -y = : ae eb) 0 


dy _ x+l 


Ex. 2. To solve 


da 4/(a? + 22 + 8) 
In this we substitute z = 2% + 2% + 3, and 
dz 
dx 
dy  dydx _ /¢ 
dz dt G2e as) Os 
den een o/h 
ce Sey L 


= 2(2 + 1) 
Therefore 


whence y= 2 +0 = 4/22 + Qe + 8) 4+ C 


CHAP. IX LOGARITHMIC FUNCTION 113 


This is a particular case of the equation 


a = (ax + b)f’(ax® + 2ba% + c) 


the solution of which is 


y = if(aw? + 2bv +c) + C 


101. The logarithmic function. 
The integration of x” in the form 


1 
-—lyn — n+l , 
Dees ne els Ons 4 


raises the question of the solution of the equation 


iO 
dz x 
The formula given in Art. 95 solves a set of equations which may 
be written dy 
Asta Lid stage Ge Reg pet Ee prt dei 


this set exhibits a gap which it is our desire to fill. As we have 
solved dy 


spe 
for all values of n, except — 1, we might have arranged a series of 
equations which would have exhibited the want of continuity in 
the sequence of indices more completely, though perhaps not so 
strikingly. The important matter is that 

dy 1 


dx «2 


an 


is the only equation of the type which cannot be solved by a mono- 
nomial in wz. 

The problem is solved by discovering a function whose differential 
coefficient answers the question proposed by the equation; the 
function sought is the logarithmic function, whose differential 
coefficient we shall now discuss. It will perhaps be of use to recall 
some of the properties of the function. 

The definition of a logarithm gives the identity 

a'8a* — 7 
where a is the base. The base is, for ordinary calculation, taken as 10, 
the basis of our system of numeration, but another base is used 
in the infinitesimal calculus, which is the incommensurable number 
e = 2-7182818..., the limit of (1 + 2)", when x->0. This number 
is discussed in Appendix II; it shares with x the distinction of having 
ince HL 


114 INFINITESIMAL CALCULUS CHAP. IX 


been studied more carefully by mathematicians than any other incom- 
mensurable number. 


T 


m-. 


ak 


Fic. 43. 


The diagram, Fig. 43, gives the graphs of log,w and log,x, which are 
respectively HH’ and TT’. The function log2 is continuous in 
[0, « ], and its elementary properties are 

log.« + logy = log.vy 
log,a — logy = log,«/y 
logia" = n log a 
log.« = wlogix 
uw = 2°30258... Deptane FN ee: 1 


102. Differential coefficient of log,x. 


Let y = log.« 
y + dy = log,(x + h) 
then dy = loga(x + h) — logyx = log,(1 + h/x) 
oy 


1 1 
LE SPiN) sles x/h 
me 7, Ball + h/x) z O8all + h/x) 


CHAP. IX LOGARITHMIC FUNCTION 115 
Writing z = h/x, we have 


opposes vt 
cary 7 08all + 2) 


Now let us give to h a sequence of values whose limit is zero ; with the 
restriction that x + his always positive, z takes a sequence of values 


Piha, ives ca. 


whose limit is zero, since 2 must be finite. 
But lim (1 + 2)!” = e, 2 ~0; therefore 


since log,x is continuous at x = e. 
And if y = log,w, we have 


dy ad 1 
Bis ere Og epates = 
It follows that et =SyN ene 


We shall in the future imply, when no base is expressed, that the 
base is e, and write always 
| 


Do = log x 


As a consequence of the work above, 


d 1 0-43429... 
9g l0B1® = Slogie = — 


The gap in the series of functions alluded to in Art. 101 is now 
filled by logw. It is the simplicity of the differential coefficient 
of this function which constitutes the real claim of e to be used as a 
base in all theoretical developments. 


103. Integration of the reciprocal of the linear function. 
We can now solve the equation 


Cat wai 
dx ax+b 
For, by substituting z = ax + b, we have 
iar ay ac.) 


hence y = “log: + © = “log (ax +6) +C 


116 INFINITESIMAL CALCULUS tt eee 


As another example, we take 
dy _ ax +b 
dx axu* + 2bx +e 
The solution of this equation can be found by writing 
z2=ax* + 2bu +c 
and we obtain y = slog (az + 2bx +0) + C 


104. Integration of certain cases of R(x). 

It is not possible at this stage to solve the problem completely ; 
at present we take the case of the integration of 
P,,(@) 
P, (a) 
where P,(x) is a quadratic function which has real factors. Now, if 
P,(x) = (w — a)(@ — 8) and the result of division gives a quotient 
Q(x) and a remainder aa + b, 


R@) = 


ax +b 
D>R@) = D+] Q(x) + 6 ye ail 
By the method of partial fractions 
ciel ae) oh 
(e — a) — 8) ear — 6 
and ax +b = A(x — 8) + Bix - a). 
Writing x = a, 6 successively, 
aa +b ap +b 
Auer BS ‘ : 


Hence D-!R(xz) = D-!Q(x) + MBS oe i 4 BD- 4 4 


= D'Q(xz) + Alog(w —- «) + i: (~ — B) 
where D-!Q(x) is known, as Q(x) is a polynomial. 
The case of « = 8 is solved by writing 


a +\b) ate =a) aa be a aw +b 

(2 — a) (x — a)? tt Sa ieee 

Cp Ot Open anal Sy 
Tica meee 7 oP ee) (x — a) 


= alog (w — a) — (ax + b)(@ —- a) 
The function R(x) can also be integrated with our present know- 
ledge even if the quadratic function cannot be resolved into linear 
factors, provided that, when 


HOP ney ee 


pu + qe tr 


) 


CHAP. IX RATIONAL FUNCTIONS LL 


we have a = 2kp, b = kq; for in this case 


p-1__% + OT eee tp-1 2px + q 
pu +qut+r pe + qu +r 


= klog (pz? + qv + 1) 


Other cases require the use of functions whose differential co- 
efficients have not yet been discussed. 


Illustrative examples. 


Ex. 1. To integrate ! 
xv — 1 
1 lat laa 
N pnt ORR ge aos 
i w—1 282-1 2xr+1 
1 1 1 1 1 
Theref —1 = ot See 9 eee 
erefore iis oy ig errata 5 a 
1 1 1 = 34 
= glog(# ~ 1) ~ lost + 1) = slog” 
Ex. 2. To integrate (x — 1) (@ + 2) 
wa + 1) 
NOT (vw — Ife +2) _, 2 A 2p) 2 
x(e + 1) x(e + 1) eel 
p-1@ = We + 2) — D-1,70 EL op-11 a 9pPp-1 1 
x(x + 1) x e+] 
= % — 2log% + 2 log (x + 1) 
2 
= a — 2log 
ma + 1 
; eS 
Ex. 3. To integrate et ee oe 
(2r-*1)2 
a3 3a — 2 
N a Dia ate ee a —~ 1) 
reenter el fe a bas eo 


3 
and D-} a 


1 
= pec maa os — 92 = = = 1 tae 
| We 5 & + 2x + 3log (x —1) — (a — 1) 


105. The value of D-1x-1, when x is negative. 

In performing certain mathematical operations upon x we have 
to remember that restrictions may be imposed upon zx by the nature 
of the operation. Thus, if # is negative, it is impossible to take its 
square root ; also log x is meaningless, unless x is positive. Hence 
when we write D-ly-1 = log x 


we imply that 2 is situated in the range [0, o |. 


118 INFINITESIMAL CALCULUS CHAP. IX 
But an answer may be given to the question D-1a~! even when 
xis negative. For in this case 
Datei los) 


here, « being negative, — x is positive. That this result is true is 
shown by writing — x = X, and differentiating, 


d d oa eee 1 

qq 108 (— %) = qx OS X- a. = x (oD i 
The whole answer to the question D-17-1 is that, when @ lies in 
[0, 0], LO bros eh feye a 


and when z lies in [ -— ©, 0], 
Dag = joe =i) 


In the examples solved above we might have entered into greater 
detail. Thus, in writing 


1 La ero 
T, : wee oan Re Saihes: 
iM 2 oes 8 Nip MS eal 
we ought, perhaps, to have added that this holds when z lies in 
either { — 0, — 1] or {1, © ], and that 
peal ae ae 


od ed td Bel eid 
when « lies in [ — I, 1}. 


106. Integration of sinx, cosx, tanx, secx, sin?x, cos?x. 
As a direct consequence of formulae of differentiation, we have 


D-tsinz = — cos x D=! cos 2 ="sin'x 
alate 1 
Again, since tanz = ——tanasecx 
sec © 
1 


he te ye Ss 
Sel EEG a Neen en 


therefore D-ttana = logsecx 


sec x tan x + sec?x 


Also, we have secx = 
sec x + tan x 


d 
t qa SEC # + tan x) 


Ue eseq met tan 
and D=-! seca = log (sec x + tan 2) 
Similarly D-tcot x = — log cosecax (= log sin 2) 
D-cosec x = — log (cosec x + cot 2) 


CHAP. IX CIRCULAR FUNCTIONS 119 


The integration of sin”x, cos™x, where m is a positive integer, 
can be affected by expanding these functions in a series of sines or 
cosines of multiple angles. Thus, by Trigonometry, we know that 

sin?a = 4(1 — cos 22) 

cos*z = 4(1 + cos 22) 

singx = 1(3sin a — sin 32) 

cos®4 = 1(3 cos x + cos 32) 

sinty = 4(3 — 4cos 2a” + cos 42) 

cosy = 1(3 + 4cos 2x + cos 42) 
whence it follows that 

D-!sin2a = d(a — dsin 2x) 

D-! cost = d(@ + $sin 22) 

D-'sin§z = 4(- 3cosa% + 4 cos 32) 

D-' cos’ = 4(3 sin x + 4sin 32) 

D-'sinta = 43% — 2sin 2x + tsin 42) 

D-'costz = 4(3% + 2sin 2x7 + } sin 42) 


EXERCISES IX 


Find the integrals of the following functions 


lol+a2 +2 2. (1 + 2)(2 — 2) a Ueeae 
CAVEAT nal heats oa 5. (a2 — a2)? 6. (1 — x)(1 + 2%, 
7. ax + bx 8. (ax + b)(cx + d) 9. 2° +a? 
10. a + 1f/s/x Ll. (@ + 1)/f/a 12. (a* + 2 + 1)/24 
3 _ 3 
iy re Te oan edhe tc 
£ Te meal) 1 — 2x 
1g, ==} 7 nay ey 
22+ 3 zg? — 37 + 2 4 — 3 
1 w2+ 2 x 
iS G1 | 9 sea ea A 
xu? — 44 + 3 a—a+1 z* — at 
"Te 
RO: ote mE, Sine PPO gas Daten Ee 
(x? + a?)? x+l 2e27 —-a - 1 
Pree eck cu! 96, 2& +1 Oy RENAE: 
6x2 + 5a — 6 x(a* — 1) x(x + 4) 
Evaluate 
28. D-1 g/x 29, D-11/23 30. D-ly/(2e + 1) 
31. D-\(2e + 1)-2 = 32. D-Naw + b)” 33. Da — 1/x)? 
34. D-Ya2 + a+ 1% 35. D-U3x—1)(e@ +1) 36. Dw + 1)%/x 


CHAP. Ix 


120 IN FINITESIMAL CALCULUS 
Ose. Dalle 1) 38. D-}(a? — 3)(a2+1)a-% 39. D(a — 1)?/4/a 
; nS Cee 
40, Dt Wo Dee 12. Doe 
x(x — 1) a(x — 1) x(x — 1) 
2 Shy us 
Yoon Bok me am Ve phe vere 45. D=t ee 
e+ 37 — 5 w+oaedt+i) w+e% 41 
Nae bey plea 4g, p-1202 + 28 
oe (xc + 1/2 (2a — 3) 
THA Dir Raed ee iN iphone 51) Doe 
x(x + 1)(” +2) a(2a — 1)? zg — ] 
2 
Spay peo Shu Dee = 64 Dt 
a | oe — 2 (e — 1)%(~ + 1) 
BD tee 56. D-la,/(a? + 22) 57.) D>) 
(7a) . a/ (a? + x?) 
58. D-42"-l(a" 4.27)? 59. pb hog x 60. D-1__! 
x x log x 
61. D-'sin 32 62.71) -136. sin? 32 63. D-l(sin 2 + cos x)? 
HG wy oe BB ysl eee 66. D-! tan2z 
1+ sina COS & 
67. D-1tanta 68. D-!tan?a 69. D-lsecta 
70. D-1sec a tan? x 7a Dee 712: Doe 
1 + tanz csinw + dcos2 
73. D-1sinmacosnzx 74. D-sinmasinnz 75. D-!cosmacos nx 


Cle wade IDI eS 


AREAS. VOLUMES 


107. First application of integration. 

The main object of this chapter is to apply integration to problems 
of mensuration. But before turning to it, a simple problem of 
kinematics is taken ; this course is adopted because the language 
of mechanics can be interpreted so directly by the symbols of the 
infinitesimal calculus. 

It has been shown in Art. 88 that, if the displacement (x) of a point 
moving along Ox is known at every instant (é), the velocity (wv) is 
deduced by differentiating 2 with respect to ¢, and that by differen- 
tiating v the acceleration (f) is determined. We shall now show 
when f is constant, (i) how v may be found and (11) how x may be 
determined from the value v; that is, we find the position at any 
tume of a particle moving along Ox with constant acceleration, when 
its initial velocity and position are given. 

With the notation of Art. 88, 


dv 
anew 


Since f is constant, we have on integration 


Again, if vy = V, when ¢ = 0, we have, C, = V, and 


da 


Integrating this, a= 4f?+Vi+C, 
Here, supposing that x = 0, when ¢ = 0, we obtain 
C, = 0 
and the well-known formula is found, 
z= +f + Vi 
We could, by taking f as a polynomial function of t, work out other 


cases, but such examples are not worth discussion, as they do not 
occur naturally. Other problems which are of importance will be 


122 INFINITESIMAL CALCULUS CHAP. X 


discussed at a Iater stage. The problem given allows the student to 
form a conception of the application of integration to problems in 
kinematics ; indeed, all such problems are reduced by laws based 
upon experiment to relations connecting acceleration, velocity 
and position, the further reduction of which is effected by the 
infinitesimal calculus. 


108. Differential coefficient of a trapezoidal area A(x). 

Let A(x) be an area bounded by the axes Oz, Oy, a curve whose 
equation is y = (x) anda variable ordinate defined by the abscissa 2, 
we proceed to determine dA 

dx 
It will be assumed in the first instance that y = ¢(x) lies wholly 
above Ox. 

Let A(x) = OMPB x=OM 
then A(x) + dA(xz) = OM'P’B 

OA) = PMP 
Now it is clear (see Fig. 44) that there is some point Q@ in PP’ 
such that, if RQR’ is drawn parallel to Ox to meet the ordinates at 
P and P’ (either being produced, if necessary) in R and R’, then 
the rectangle RM M’'R’ is equal to the area PM M’P’; hence 
dA(x) = rect. QN . MM’ 

Now let ON = a + 06x, where @ is a proper 7 
fraction. Then we have 

OA(x) = dx o(x + 062) B 

OA (a 0 M NM 

a p(x e 0 dz) Fic. 44. 

and the relation holds even when oz is negative. Proceeding to 
make dx > 0, o(a + 9dx) — (x), and we have 


dA 
Fix, = (x) 


It is assumed in this proof that the curve y = ¢9(#) is continuous. 


109. Area bounded by a curve, the axes and an ordinate. 
The restrictions imposed in Art. 108 hold in the case of the curved 
boundary of this area. We have 
dA 
dg 7 9 
If we can find a function which when integrated gives (x), we 
have asolution. The constant in the solution has to be determined ; 
this is found by noticing that 
A(0) = 0 


CHAP, X INTEGRAPH 123 


110. Integraph. 

The solution of the problem in Art. 109 is effected practically by 
an instrument called the Integraph, the principle of which is directly 
deduced from considerations already proved. 

Let BP be the curve y = g(x), and let MK be measured to the 
left along Ox from M, the foot of the ordinate of P, and be of 
unit length ; also let Mp be equal to the area 
OMPB | = A(x]. Then 


tan MKP = MP/KM = MP = o(2) 


Again, tan Mtp = = = MP = (2) 
Therefore, KP is parallel to the tangent at p. 
The Integraph is designed with two pointers 
P and p so controlled that as P describes BP, 
p moves so that the tangent to its path is parallel to KP. In such 
an instrument the reading given by Mp in any position is the area 
OM PB, provided that the pointer starts at O, and therefore A(Q) = 0. 


Fig. 45. 


111. Illustrative examples. 


Ex. 1. To find the area of a right-angled triangle. 

Let OBC be the triangle; take the axis Ox along OB and consider 
a point P on OC. 

The equation of OC is y = max and A(z) is the 


area OMP. C 
Then nS = VP = mx E 
dx 
A(x) = 3mx? + C, uO: M eee 
and since A(0) = 0 C, = 0 Fic. 46. 
whence A(x) = 4m2? 


The area OBC = A(b) = 3mb? = 34mb.b = 30B. BC. 


Ex. 2. To find the area of a trapezium. 


Let Ox be perpendicular to the parallel sides, and let the equation of 
BC be y= max +6. Then, taking A(z) as the 
area OMPB, we have 

dA 
Bae aie + 6 
A(x) = ma? + bx + C, 
Also C, = 0, because A(0) = 0, and 

A(x) = 4u(mx + b + 6) = 430M(BO + PM). 
Thus, theareaOBCD = 40D(BO + CD). 
Again, the area OB’C’D = 40D(OB’ + DC’). 
Therefore the area of the trapezium = 40D(BB’ + CC’). 


Fic. 47. 


124 INFINITESIMAL CALCULUS CHAP. X 


Ex. 3. To find the area of the part of the curve y = x?(3 — a) which lies 
in the first quadrant. 

We require the area OAPO, but we must solve the more general 
problem of finding the area OMP = A(z). 

p 

oe = MP = 327 — 23 

dx 
A(z) = 
no constant being added, because A(0) = 0. 

The area required = A(3) = 27(1 — 3) = 63. 

This answer may be checked by drawing 
the curve on squared paper and estimating 
from the diagram the number of squares and Fig ae 
portions of squares included in the area. 


Now 


The examples just given illustrate the application of integration, 
(i) to two problems whose results are well known, and (11) to a 
problem which might be insoluble by elementary methods. This is 
characteristic of the power of the method. 

It may be noticed, too, that, whenever we know the solution of 
an equation dy 
ie p(x) 


we have the solution of a class of problems concerned with the area 
bounded by y = (x), the axes and an ordinate. 

Further, in each of the examples a far more general problem is 
solved than the single particular instance presented for solution; this 
was especially the case in Ex. 3. Indeed, the solution of 


dA 
eae o(x) 


when we know it solves a second class of problems in which the 
constant C has different values. For, if we take A(x) as the area 
(supposed to be above Ox) lying between y = (x), Ox and two 
ordinates, one of which is fixed, we have still 


dA 
eine p(x) 


and by integration we obtain A(x). Thus, in 
the diagram, if the area BCC’B’ is required, 
we take = A(z) = area BOMP. 

In the solution of this problem we deter- 


mine the constant by writing A(b) = 0; the required area is then 
A(b’), where 6 and 0’ are the abscissae of B and B’ respectively. 


Fic. 49, 


Ex. 4. To find the area of 
y = (x + 2)(1 — 2) 
cut off by Ox. 


CHAP. X SIGN OF AREAS 125 


The parabola cuts Ox at A(1, 0) and B(- 2,0). Let the area 
BMP be A(a), then 


= =y=2—2 — x 
A(x) = 2a — 4a? — 403 + C P 
Now A(— 2) = 0, therefore C = 34, and. 
A(x) = 2x — 4x? — 1a? + 3h ) Oo MiAm 
Hence A(1), the required area, is 44. Fic, 50. 


112. Extended meaning of A(x). 

In Art. 108 we found A(z) on the assumption that y = (x) was 
wholly above Ox; we must now discuss the new meaning which 
may be given to A(x) in the case in which y = (x) crosses Oz. 

If we regard o(x) as the derived function of A(x), it is clear from 
Art. 42 that so long as (x) is positive, A(x) increases, but that, 
when (x) is negative, A(x) decreases. 

Again, referring to the integraph curve (Fig. 51), in which Ce 
is equal to the area OCB, we see that after passing C, the ordinate 


Brey 51: 


of the curve, y = o(x) becomes negative, the pointer of the integraph 
which describes the graph of A(x) approaches Ox. To estimate the 
fall, let us for a moment regard y = g(x) in the sense in which 
Oy’ is the positive direction; in this case, the fall becomes a rise 
and the rise from c to p is equal to the area CPM. Returning 
now to the usual conventions, it follows that 


Mp = Cc - (Cc —- Mp) 
= area OCB — area CPM 


Also looking at the diagram in which y = A(x) meets Ow at d, we 
see that the area OCB = the area CDd. 
Thus if A(z) is the solution of 


dA 
AR = (x) 


even though y = (x) may cut Oz, we have a meaning for A(q) ; 


126 INFINITESIMAL CALCULUS CHAP. X 


for, denoting the trapezoidal areas given by y = (x), and Ox by 
A,, Ag, A;,..., of which A, is above Oz, A, is below, and so on, we have 
A(x) —— A, ais A, + A, <TTh leaie 

The reader may have noticed that the areas have been throughout 
lettered in the order in which they would be described in a counter- 
clockwise circuit. Just as an athlete runs round a track with his 
left foot always inside, so we have suggested a definite direction 
of description. Again, so long as the area is above Oz, our con- 
vention has agreed with another convention made in 


ak 
Pe 


namely, that the part of the axis of # included in the circuit is also 
described in the direction of w increasing. Now both of these 


Fic, 52a. Fic. 520. 


requirements are satisfied if we suppose that the boundary of A(z) 
is described so that the axis of x is described in the direction Ox and 
that the contribution to A(#) is positive when the area described 
lies on the left, and negative when the area is on the right. The 
first figure (Fig. 52a) illustrates the two cases separately ; in the 
second (Fig. 52b) the two cases are combined. The function A(z) 
in Fig. 526 is then 


the area OCMPDECBO = OCB -—- CDE + DMP 


113. New interpretation of the relation between f(x) and f’(x). 

With the extended interpretation of A(x) we have a complete 
interpretation of the relation between the functions A(x), (a), 
which are connected by the equation 


dA 
But we have already discussed the same relation with different 
symbols.in the form d 


The two problems may be considered together by writing 
A(x) =f) — 9(@) = f'@) | 


CHAP. X VOLUME BY INTEGRATION 127 


and a new interpretation of the results of Art. 46 obtained by 
considering f’(x) as the original graph and f(«#) as a function which 
expresses the area (in its extended meaning) lying between Oz, 
the curve y = f’(x) and the ordinate given by the abscissa a. 


114. Volumes of solids of revolution. 

Let V(x) be the volume generated by the revolution about Oz 
of the area which in Art. 108 was denoted by A(x). Then, referring 
to Fig. 44 and taking V(x) + dV(x) as the volume generated by 
OM'P’B, we have 


O6V(x) = volume generated by PM M’P’ 


= MM’. QN? 


where Q is some point of PP’, but not in general the same as the 
point in Art. 108. 


Hence aw = TQN? = xr[o(x + dz) }* 
where @ is a proper fraction. The value of 0 varies with the length 
of dx, but is always between 0 and 1. Proceeding to the limit 
in which 6x — 0, we have 


I 


d 
oh = t[(x) |? 


The problem is simpler than the problem of areas, inasmuch as 
[(x)]* is essentially positive and V(x) does not decrease even 
when y = (x) crosses the axis. The extended meaning when 
the axis is crossed being that V(x) is the sum of the volumes 
generated. 


The student should notice that the integraph can be applied to 
this problem. For, if we write 


£ (Vim) = fel)? = x02) 


and trace the graph of y(x) on the paper, then the integraph when 
applied to y = y(x) gives us the quantity V/z. 


We may notice that the solution of 


dy 
Fie eth 


when f(x) is positive, provides us with the solution of two-problems, 
(i) the area bounded by the graph of f(x), two ordinates and Oz, and 
(ii) the fraction 0-31831... of the volume generated by the revolution 


round Ox of the area bounded by the graph of [ F(x) }, two ordinates 
and Ox, 


128 INFINITESIMAL CALCULUS cmap. x 


115. Illustrative examples. 
Ex. 1. To find the volume of the frustum of a cone. 
The diagram is a section through the axis of the cone, which is Oz. 
The equation of the upper generator is 
y=r+(R-rjaf/h (Fig. 53) 
where x = 0, x = A give the ends of the frustum 


NAGE Ah et al 
ee Ebay 17 Tr h «| 


hr 
C iT ce 
h R-r 3 hr 
Whence hh eee Mad Lie ga Rest 
WAS ie droga Jr png | UT eas os 
And the volume of the frustum = V(h) 
= drone = (Re — r?) = 4nh(R* + Rr + 7?) 


4 


a 
' 

1 
R 
\ 
\ 
( 


A MoO eM B 
RiGee: Fic. 54, 


Ex. 2. To find the volume of the frustum of a sphere. 
Let the equation of the circle which generates the sphere be 


Then hs = my? = n(r2 — 2) 
V(x) = n(r2x — du) + C (Fig. 54) 
Now, if V(x) is the volume generated by AMP, then when x = — r; 


V(x) = 0, whence C = énr°, 
V(a) = r(r?2a — 403 + 273) 
= An(r + x)(2r? + rx — x?) 
As verifications we have the volume of the hemisphere obtained by 
writing « = 0, and that of the sphere by writing x = r. The volume 


of the frustum generated by MBP is equal to the symmetrical frustum 
generated by AM’P’, which is obtained by writing — a for x. 


CHAP. X PARABOLIC APPROXIMATION 129 


Ex. 3. To find the volume of a paraboloid. 
The generating parabola is y? = 4az. 


d 
We have = my* = 4naxr 
V (30) et Dra" 


no constant being added, because V(0)= 0. The answer is usually 
expressed by saying that the volume of the paraboloid is one-half the 
volume of the cylinder upon the same base and of the same altitude. 


116. Approximative equation of a curve given by three points. 

A single observation of a physical quantity which is a function of 
x gives one point upon its graph; m observations provide us with 
n points. Now, from the points available it is often possible to 
conjecture the shape of the graph in the neighbourhood of these 
points ; the more points there are, the more plausible is the guess. 
If we have only two points, we assume a linear graph and join the 
points by, a straight line; if there are three points, we assume that 
the function is quadratic and draw a parabola through the given 
points, having its axis perpendicular to the axis of x. 

Let us take the case of three observations and suppose that they 
are equidistant ; then, if we choose our axes 
properly, we maysuppose that the three points 
upon the graph are 


( a h, Y) (0, Yo) (h, Y3) 
Assuming that the function is 


y=at+ be + cx 


we have 
y, =a — bh + ch? Yo =a Y; =a + bh + ch? 
It follows that 
Y, + Ys = 2(a + ch?) Ys — Yy = 2bh 


and that 
ie EUs at I see 
LES pe ey a Dh 
The form of the function is given by 
Y3 — VANS 2 Y, — 2Y, + Ys 2 


ELIE TOTO Oh? 


The reader must understand that the assumption of a quadratic 
function as the form of the function is only one out of many which 
could be made. Its artificiality can be illustrated by taking a 
second form which will be useful afterwards, namely, 


y? =a + bu + cr? 
C.C. I 


130 INFINITESIMAL CALCULUS CHAP. X 


Here, the geometrical assumption is that the curve which is the 
graph of the function is a central conic, having one of its principal 
axes upon Ox and passing through the three points. Introducing 
the same data into this equation, we have, by an almost identical 
series of algebraical steps, 


as, 2 » STP 2 2 
yon ye + Wg 4 da ea 


The two functions thus found are both useful: the first will be used 
to find an approximative expression for the area bounded by the 
graph, Ox and the extreme ordinates, and the second to find an 
approximation to the volume generated by the revolution of this 
trapezoidal area about Oz. 


117. Simpson’s rule for a trapezoidal area. 


If we take an area bounded by a curve, two ordinates and the 
axis, Simpson’s rule provides an approximative formula for this 
area when the extreme ordinates, the intermediate ordinate and the 
width of the area are given, provided the curve lies on one side of Ox 
and passes continuously from one end to the other. 

For if we indicate the three ordinates as yj, Y5, y3, in their order 
of sequence, we may assume that the curve is a parabola through the 
three points with its axis parallel to Oy and write as an approximate 
form of the curve the equation found in Art. 116, 


Yo ~ Ye , Ya — Yo + Yous 


ier LA Nec? Dh2 
Then, since 
Cae Y3 -— YW Yi, — 2Yo + Ys 6 
dene wea) 5a he 
In le by sag 


Also, remembering that A(- h) = 0, 


mab Abo chbaahi ae: = 242 + Ysn5 
C = Y th +h 6] h 
and the area enclosed by the extreme ordinates = A(h) 
SR re ty ta Pe A one 


Zyoh + EY, — Be + Ys)h 
3(Y1 + 442 + Ys)h 


A couple of simple tests of cases in which the formula is exact may 
be given; they will perhaps even suffice to enable the student to 


CHAP. X SIMPSON’S RULES 131 


write down the formula, if he recollects its general form. Thus, in 
the rectangle y, = y, = y3 and the area = 2y,h; in the trapezium 
2Y2 = ¥, + yz, and the area = 2y,h. The formula gives, of course, an 
exact result also in the case of the parabola. 


118, Extension of Simpson’s rule. 

It is not difficult to extend the formula to meet the cases in which 
more than three equidistant ordinates are given, and also when 
the ordinates are not equidistant. But it is usually sufficient to 
tackle such problems by successive applications of Simpson’s 
tule as stated above. 

Thus, if seven equidistant ordinates are given, the area is divided 
into three sub-areas by y, and y;, and Simpson’s rule is applied to 
them. The approximation to the area obtained is 


$Y, + 44, + Ys + Ys + 44, + Ys + Ys + 446 + yh 

= 31% +4 +42 + Ys + Yo) + 2; + ys)IA 
from which a rule is at once contrived to meet the case when the 
number of given ordinates is odd ; when this number is even, it is 


best to take the area bounded by the first or last pair separately, and 
to treat the remaining part of the area by the above method. 


119. Simpson’s rule for volumes of revolution. 


We suppose as before that in the generating area three equidistant 
ordinates are given, and we assume that the generating curve has 
for its equation 


Cy eee: 2 _ Oy 2 2 
yz = y2 + 23 wo +B Ap + Ys" 2 
and solve zat = ny? 


Proceeding as in the case of the trapezoidal area, 
V = gn(y? + dy? + y,")h 

Simple tests corresponding to those given in Art 117 are numerous : 
the cylinder in which y, = y, = y, = r and whose volume is rr?. 2h; 
the cone in which y, = 0, 2y, = ys = r and whose volume is irr”. 2h; the 
sphere in which y, = y, = 0, y, = r = h, whose volume = $m; and 
the frustum of the cone in which y, = 7, yz = R, y, = 3(r + RB), whose 
volume = in(r? + rR + R?*)2h. 

These results are accurate because the equation of the generating 
curve is in all cases expressible in the form 


y2=a + ba + ca? 


120. Trapezoidal areas whose bases rest upon Oy. 


We now discuss the area between a curve which does not cross 
Oy, the axis of y and two lines parallel to Oz. 


132 INFINITESIMAL CALCULUS CHAP. X 


The area required is indicated in the diagram; it is the area 
B,BCC,. The solution may be obtained from the previous work, 
for we have | 

area B,BCC, = rect. OC —- rect.OB — area BB’C'C 


The same problem is solved directly 
if we can express the equation of the 
curve in the form 

= xy) 
For then, if we write the area B,BPP, as 
Ai(y), we have, just as in Art. 108, 
dA,y) _ | 
dy yAC) 


and by integration we can obtain A,(y). 


Volumes generated by the revolution of B,BCC, about Oy are 
determined by solving the equation 


Vi 


d 
dy = tly (y)]? 


121. Areas of certain oval curves and the volumes generated by their 
revolution. 


Consider the curve whose equation is 


y = f(x) + g(a) 
where g(b) = 0, g(c) = 0 and g(a) is defined only for a range (6, c) of a. 
The dotted line y = f(x) on the figure bisects the chords of the oval 
which are parallel to Oy. 


Let PQ be one of these chords, so that 
y, = MP = f(x) + g(x) 
Y2 = MQ = f(x) - g() 
and let A(w) be the area BQPB; then we have 
that dA(x) is the area PQQ’P’ = QP. dx 
nearly. Proceeding, we find 
ee = QP = MP — MQ = 29(z) Fi. 57. 
By integration A(x) is found, the constant is determined by writing 
A(b) = 0, and the whole area is then given by A(c). 
The application of this method to examples leads generally 
to problems of integration of which we have not at present 
treated. 


The volumes generated by the revolution of oval curves of this 
type which do not meet Ox may be obtained in the same way. If 


CHAP. X EXERCISES 133 


V(x) is the volume generated by the revolution of A(x) about Oz, 
we find d 


V 
age TY? — Yo") = Aref (x) . g(x) 


The integraph is applied very simply to such problems as the 
determination of the oval area by allowing the pointer to travel 
round the whole circumference, BQCPB. For the area BB’C'CQ 
will be recorded as the abscissa increases, while CPBB’C'’ with a 
negative sign is recorded as the abscissa decreases, the pointer 
moving from C to P to B. Then the total area recorded will be 
the difference between the numerical values of the two trapezoidal 
areas, which is the area of the oval BQCP. 


EXERCISES X (a) 
Areas 


1. Show (i) that the area bounded by y = 27, x = 1, y = Ois4, and 
(Wyeluar the areca pounded by y*\ = a7 Ja? = Ly 0 is 2. 

2. Prove that the axis of x cuts off from y = 2v — 3x7 an area equal 
to +. 

3. Show that the area of y = (x + 1)(@ — 2) cut off by Oz is 44. 


4, Find the area of the segment of y? = 4ax cut off by x = a, and 
prove that it is bisected by # = 4a./2. 


5. The curve y = x(x — 1)(x —.2) cuts the axis at O, A, B, in order. 
Show that the areas subtended by OA and AB are equal. 


6. Show that the area included between Oz and a semi-undulation 
of y = sin x contains 2 units. 


7. Prove that y = sin? # bisects the area of the rectangle rare by 
the axes, v = $7, y = 1. 


8. Show that a”y™ = bx” divides the rectangle formed by the axes, 
x = a, y = 6 into two parts whose areas are in the ratio of m: n. 


9, Find the area bounded by the curve 6zy = z+ + 3, the axis of x 


aud the ordinates ¢ = 1,2 = 2: Ans. 3 + 4 log 2 = 0:972.... 
10. Find the area bounded by (1 — 2*)y = (x — 2)(a — 3), the axis 
of x and the ordinates x = 2, x = 3. Ans. log (241 73-5) — "1. 
11. Find the area bounded by a’y(w# + a) = at,x2=0,y=0,2% =a. 
Ans. a* (log 2 — 5%). 

12. Find the area between (x — a)*y¥=a2(x +a), y=ax + 8a, 
aii) ane, Ans. a*(5 log 2 — 1). 


13. Show that the area between y? = 4ax and the tangents at the ends 
of the latus rectum is 4a?/3. 


14. Show that the area between y = sina, y = sin 2% and a = an 


is }. 


134 INFINITESIMAL CALCULUS CHAP. X 
15. Find the area between y = axz/4/(x* + a’), y= a, © = 0, a = as 
: Ans. (2 — 4/2)a?. 
16. Find the area of 5y? = 2a(% + a) cut off by the double ordinate 
x = 3a/5. Ans. 128a?/75. 
17. Find the area between y?(a — x) = a andaw=a. Ans. 3na*/4. 
18. Show that the loop of ay? = «?(a — x) encloses an area equal to 
8a?/15. 
19. Prove that the area enclosed by the loop of c®y? = «?(a? — 2?) 
is 2a3/c. 
20. Prove that the ordinate x = a divides the area between 
af 20 ft A ee ee 
in the ratio of 3x — 8: 3n + 8. 
21. Show that the whole area of the curve given by 
2 = acos*@ y = asin? 
is 3ra?/8. 
22. Trace the curve 2x”? — 2ay + y? = 4, and show that its area is 47. 


23. Show that y = + a/(1 - 2) isa curve resembling a figure of 
eight, and that the area of each loop is 2. 


24. Show that x = 4a cuts off from the parabola (2% — y)? = 4ax 
a segment whose area is 6407/3. 


EXERCISES X (B) 
Volumes 


1. Show that a sphere of radius 1 foot is divided by a plane distant 
4 in. from the centre into two parts whose volumes are in the ratio 
OLa7: 20. 


2. Prove that the volume of the segment of a sphere whose height is 
h and the radius of whose base is ¢ is 4nh(3c* + h?). 


3. Find the volume of the spheroid generated by the revolution of 
the ellipse a? + 4y? = 16 about its major axis. Ans, 647/3. 


4, The segment of the parabola y* = 4ax lying between x = a, 
x = 2ais revolved about Ox. Show that the volume generated is 6za?. 


5. Find the volume generated by the revolution about Ow of the 
areas defined in Question 1, Exercises X (A). Ans. £0, tT. 


6. Prove that the volumes generated by the revolution about Oz of 
the loops of the two curves 
yr =ae-1j\(e@-2) wy = a(l — x(x — 2) 
are equal. 


7. The area bounded by a*y = x(x — a), the axes and x =a is 
revolved about Ox. Show that the volume generated is ma3/105. 


CHAP. X EXERCISES 135 


8. The area bounded by y2x = 4a%(2a — x) and the ordinates 
x = a,x = 2a is revolved about Ox. Show that the volume generated 
is 4ra3(2 log 2 — 1). 


9. Show that the volume generated by the revolution about the 
latus rectum of the segment of the parabola (latus rectum = 4a) cut 
off by its latus rectum is 327a3/15. 


10. The area bounded by a? = y2(2 — y), the axes and y = 2 is 
revolved about Oy. Show that the volume generated is 47/3. 


11. Prove that the volume generated by the revolution of 
(aa a")2 =) 2a°s 
about Oz is na’, 
12. Show that the volume generated by the revolution of 
a" /a* + y/o? = 1 
about the tangent at the end of the major axis-is 27?a?b. 


13. A cylindrical hole of length 2c is bored centrically through a 
sphere. Prove that the volume of the part left is 47c?/3. 


14. Prove that the volume generated by revolving about its axis the 
segment of a parabola cut off by the latus rectum bears to the volume 
generated by revolving the same area about the latus rectum the ratio 
of 15: 16. 


EXERCISES X (c) 
Simpson’s Rules 


1. If Simpson’s rule is applied to the calculation of the areas in 
Question 1, Exercises X(aA), show that in the first case the correct answer 
is found, and that in the second case the error is about 0-5 per cent. 


2. If Simpson’s rule is applied to the calculation of the volumes in 
Question 1, Exercises X (B), show that the correct answer is given. 


3. Apply Simpson’s rule to determine the area between y = cos 2, 
y = 0, « = 0, x = 4n. Show that the answer is in excess by about 


cam lOz. 
4, If the trapezoidal area defined by the curve 


be xr+2 
Yo a(o +.1)(40— 2) 
the axis of x, and (i)a = — 4,2 = — 2; (li)w = 0,4 = 3; (ili) x = 5, 


x = 7 be denoted with positive signs by A,, dy, Ag respectively, prove 
by integration that the values of A,, Ay, Ag are 0-125..., 1:94..., 1-261... 
respectively, and by Simpson’s approximative rule are 0:123..., 1-99... , 
1-276... « 


5. By applying Simpson’s rule to calculate the area bounded by 
sy =1,y = 0, « = 1, x = 2, find an approximate value of log 2. 


136 3 INFINITESIMAL CALCULUS CHAP. X 


6. Show that Simpson’s rule is correct when applied to a curve 
y = f(x), when f(x) = 1, x, x*, x, and also when 
f(z) =a + bx + ca® + daz? 
7. If y1, Yo, Ys Yq are four equidistant positive ordinates of the curve 
y=at bx + ca*® + da 
given by x = 0, h, 2h, 3h, show that the area between the curve, the 
axes and « = 3h is 
B(Y1 + 3Y¥_ + 3Yyg + Yah 
8. If y1, Ya, Ys are ordinates of a curve which lies above Oz, and if 
the distances between the consecutive pairs of ordinates are a, b, prove 
that the area bounded by the curve, Ox and the extreme ordinates is 
approximately 
a+b 
-6ab- 
Find the corresponding formula for the volume generated by the 
revolution of this area about Oz. 


{(2a — b)by, + (a + b)?y_ + (2b '— a)ays} 


9. If7,, ro, 7, are the three radii of the curve 
r=a + .b0 + c0?+d068 
defined by the angles 8, 8 + a, B + 2a, show that the sectorial area 
bounded by 7,, r, and the curve is 
L(y? + 4ro? + 15?) 
10. Given that y,, Ys, y, are the radii of equidistant cross-sections of a 


surface of revolution, prove that the moment of inertia about its axis 
of the volume bounded by the first and last section is apprcximately 


dn(yy* + 4yo* + yg*)h 


CHART Re xX] 
MOMENTS BY INTEGRATION 


122. The moments of a system of particles distributed along Ox. 
Let m, be at x, m, at 2p, ..., then the first moment of the system 
about O is Net iMekait .. = LIne Na 
Again, the second moment, quadratic moment or moment of inertia 
about O is MX" + Mo? +... = Lmx? = M, 
The mass of the system is 2m. 


123. First and second moments of a line-distribution of matter. 
Let the matter be distributed along AB, where x = a, x = 6 give 

the coordinates of the terminal points of the distribution, and let 

the density at P be o(x) or 9, as we may 

sometimes call it. FQ 


Now, if e is constant, the mass of the 
element PQ (= 6x) is p dx, but if p varies 
and is continuous there is some point x + 0 dx in PQ at which the 
density o(~ + 06x), when multiplied by dz, gives the mass of PQ.* 
Hence the total mass is Lo(* + Odx)dx. 


Again, if we take the line AB and cut it up into lengths equal to 
6x, we can find points in each small section at which the mass of 
the element must be placed to give a system of a finite number 
of masses which has the same first moment about O as the 
given continuous line-distribution. Let the point of PQ at which 
o(a + Odx)dx is placed be x + 0’dx, where 0, 0’ are both positive 
proper fractions. The first moment of PQ about O is now 

o(a + Odx) . (x + 0'dx)dx 

Also, if M,(x) is the first moment of the matter along AP about 

O, then M,(x + 62) is the first moment of AQ, and 
OM,(x) = first moment of PQ 


= o(x + Odx) . (a + 0’dx)dx 
ee. = (x + Odx) . (x + 06x) 


* The reader may compare the argument used in Art. 108. 


Fic. 58, 


whence 


138 INFINITESIMAL CALCULUS CHAP. XI 
Proceeding to the limit in which 6a + 0, we have 


M, 
em o(x) . 2 


Again, if we take M(x) as the second moment of AP about Oz, 
we find by a similar argument that 
OM,(x) = o(@ + Odx) . (x + 0" dx)? 0x 
where 0 is the same as before, but 0” is another proper fraction which 
is not in general the same as 0’. Whence we obtain 
dM 
1 = 0 (2) ue 
If m(x) is the mass of AP, we also have 


d 
a. = e(%) 


These three differential coefficients of M,, M,, m allow us, by 
integration and determination of the constants which occur, to find 
the first and second moments and the mass of AP, the constants 
being determined by 

M,(a) = 0 M,(a) = 0 m(a) =0 


The first moment of AB is now M,(b), its second moment M,(b) 
and its mass m(b). 


124. Illustrative examples. 


Ex. 1. To find the first and second moments of a rod of uniform section 
about a point O, distant a from one end, O lying in the line of the rod 
produced, 


Let OA = a,OB = bgivethe ends; oe being constant. Wehave 
dM, 
dx 
M,(x) = $ou? + C 
and M,(a) = 0; therefore 
M,(x) = 39(a? — a?) 
and M,(b) = $9(b? — a?) = 4m(a + 5) 


when m is the mass of the rod. 


= ex 


Again, for the second moment we have 
dM, 
da 
M,(x) = tea? + C 
= $0(x* — a®) 
M,(b) = 4m(b? + ba + a?) 


— ox? 


CHAP. XI LAMINAR MOMENTS 139 
Ex. 2. To find the first and second moments about one end of a rod whose 
density varies as the square of the distance from that end. 


This is the case of a conical rod tapering to an end, which is taken 
aS OFigin-s 9 = kx. 


dM, 

dx 
M, (7) = }kx*, since M,(0) = 0 
M,(1) = hl! 


S25 ey Sines 


Again, if m(a) is the mass of a length a, 


dm 
Se ree Lee 
aie kx 
m(x) = tka? 
and m, the mass of the rod, = 4k 
whence M,(l) = 2ml 
; dM 
Again, hese ye Lond Ge 
gal a ex ka 


M,(x) = 5ka° 
M,(t) = 4kP = 2mi? 
Ex. 3. To find the second moment of a tapering conical rod about the 
thick end. 


Here we take p = kav? as in Ex. 2, and to get the second moment 
of the element at P we multiply by (1 — x)”. Thus 


a = kaX(l — x2)? = k(l?a2 — Qia3 + x4) 


M,(x) = k(4l?23 — dla + 42°), since M,(0) = 0 


125, First and second moments of a lamina. 

The discussion of the moments of a plane distribution of matter 
about an axis Ox (or Oy) will now be sketched. In its general form 
the problem is a double summation. The area is divided into a 
large number of small rectangles whose sides are parallel to the 
axes and equal to 6x and dy in length. Ifo is the density at (z, y), 
the mass of the elementary rectangle, in which (a, y) is situated, is 
taken as cdxdy; its first moment about Oy is pxdxdy. The first 
moment of the whole lamina about Oy is the limit of the sum of all 
such elements as oxdxdy, when the number of the rectangles is 
increased indefinitely, their length and breadth being decreased 
indefinitely. Under the same circumstances, the mass of the 


140 INFINITESIMAL CALCULUS CHAP; XI 


lamina is the limit of the sum of the terms cdxvdy. Also its second 
moment about Oy is the limit of the sum of ox?da dy. 


We may simplify the discussion by considering an area lying upon 
one side of Oz which is bounded by y = f(x), Oz and a couple of 
ordinates and by taking the case in which o is either constant or a 
function of x only. 

Thus, taking M,(x) as the first moment of AA’P’P about Oy, 
we have that M,(x + 62) is the first moment 
of AA’Q’Q about Oy ; therefore 


6M,(x) = My(x + 6x) — M,(a) 
= first moment about Oy of PP’Q’Q 
= oy 0x . x approximately 
and y = f(x). Whence 


dM, _ eye Ol CAN REE 
dx AO . y Fic. 59. 


B ~ 


Again, taking M,(%) as the second moment, we deduce that 


dM, 
da 
The above method applies to the case of the standard figure given 
in Fig.59. But for sucha figure the process of determining moments 
about Ox would not be so simple, even in the case of o constant. 
The areas to which we shall usually apply our method are of the 
nature of the ellipse in which the perpendiculars from P upon Oz, 
Oy both lie in the area. For such areas we shall introduce the 
symbols N,(y), N.(y) to denote the first and second moments about 
Ox of areas lying between two parallels to Ox, the nearer of which 
is fixed and the further is given by y. In this case we have, if o is 
constant or a function of y, that 
dN dN 
aa = OLY =H) Efe Mee 
where x = y(y) is the equation of the bounding curve. 
The four formulae 


Ceol Lah al) 


dM, _ aM oly Ne 
yom oxf(x) Fiat ase f(x) 
dN, dN, 


= 5) -gpens 2 
dy = YL) “Gye = OLD) 


(where y = f(x), x = y(y) are the same functional relation), apply 
only when o is constant. We may add that in the case of constant 
density, m (the total mass) = o . area. 


CHAP. XI AXIAL MOMENTS 141 


126. Illustrative examples. 
Ex. 1. The first and second moments of a uniform rectangular lamina 
about adjacent sides. 


Let a, 6 be the length and breadth and let c = 1, then 


dM, 
Fant rit 
| M,(x) = 4b2?, since M,(0) = 0 
Thus M,(a) = 4ab = dma 
Similarly N,(b) = 3mb 
dN 
A 2 = ay? 
gain ay ay 


N,(b) = 4ab® = 4mb? 
Similarly M,(a) = 4ma* 
Ex. 2. The first moment about a bounding radius of the quadrant of a 


circle. 
As before, we take o = 1, and the equation of the circle as 


y = 1/(a@ — x) 
Hence a yarn Aa ar) 
M(x) = C — }(a* - x2)? 
Now MLO) = O; therefore C = 4a% 
M,(x) = 4a? — (a? — af)! 
and M,(a) = 4a? = 4ma/3n 


The equation to which the expression for the second moment leads 
cannot be integrated without the use of circular functions. The 
equation is 

dM 
me = xy = x/(a? — 2?) 


The solution (which is given in Chap. XIII) is 
M,(x) = fatsin“\(a/a) + $a(2u — a*)/(a? — 2?) 


From this M,(a) = ;',na* = 4ma? 


127. Second, or axial, moment of a lamina about an axis perpen- 
dicular to its plane. 


If the axes of reference of the Jamina intersect at the point at 
which the perpendicular axis meets it, then the required second 
moment is the lim Yor?éa dy = lim Yow? du dy + lim Loy?dx dy (for 
v2 = 4% + y*), and this is the sum of the second moments of the 
lamina about two axes intersecting at right angles in the point 
at which the axis perpendicular to the lamina meets its plane. 


* 


142 INFINITESIMAL CALCULUS CHAP. XI 


It is an important deduction from the result given in Art. 126, 
Ex. 2, that the second moment of a uniform circular lamina about 
its axis is 

M,(a) + N.(a) = 4ma? + ima? = 3ma? 

This result is of such importance that it may be proved inde- 
pendently of integrations which have not yet been performed in the 
following way. Let I(r) be the second moment of a circular lamina 
of radius r about its axis, then I(r + 6r) is that of a circle of radius 
(r + Or); therefore I(r + dr) — I(r) = 61(r) is the second moment 
of an annulus bounded by circles of radii r, r + dr. But this second 
moment is equal to the mass of the annulus multiplied by the square 


of a mean radius. 

Ol(r) = 2xo(r + 0,67) dr(r + 86r)? 
dl 
dr 

I = 4nor4, since I(0) = 0 
I(a) = 4noa* = ime 
It follows from Art. 126, Ex. 1, and from the statement at the 
beginning of the article, that the second moment ofa uniform 
rectangular lamina whose sides are 2a, 2b about an axis through 
its centre perpendicular to its plane is 
4ab(a? + b%)o = 4m(a? + 6) 


also, the second moment of a square lamina (edge = 2a) about its 
central axis is 


that is = 2ror? 


2ma2 
ma 


128. Second, or axial, moments of volumes about an axis of symmetry. 


First, we take volumes bounded by surfaces of revolution and 
write I(x) for the second moment about Ox of the volume generated 
by A’P’PA (see Fig. 59). Then we have 


Ol(~) = second moment of the volume generated 
by the revolution of PP’Q’Q 


= dny'*dx 
dl 
ae ary? 
By integration I(x) is found, I(a) being zero, and if the end-planes 
are given by x= a, x = Jb, the required second moment is I(6). 

As a second problem, we take a cuboid whose edges are 2a, 26, 2c, 
and find its second moment about the c-axis. In this case, by the 
expression in Art. 127 for the second moment of a rectangular lamina, 
we deduce that dI(a) = 4ab(a? + 6%) dx 

I(x) = 4abz(a? + 6?) 


CHAP. XI CENTRE OF GRAVITY 143 


if x is measured from a face 
I(2c) = Sabc(a* + 6?) 


= 4(a? + b?) x volume 


I 


129. Illustrative examples. 
Ex. 1. To find the second moment of a cone about its axis. 


We have # = dry! = drevttanta 
x 
iia) = 1a? tant oe 
I(h) = pork? tanta = {3 r°V 


Ex. 2. To find the second moment of a hemisphere about tts axis. 


I 
= $r(at — 2a%x? + at) 
I(a) = $r(atx — 2072? + 42°), since 1(0) = 0 
I(a) = 4ra°/15 = 2a2V 
Ex. 3. To find the second moment of a pyramid on a square base about 
the axis through the vertex perpendicular to the base. 


Let 2y be the side of a square section at a distance x from the vertex, 
then AL 
in 


yt = Satat/h 


ele 


130. Centroid or centre of gravity. 
In the case of a mass-distribution certain quantities called first 
moments have been investigated ; if we take the case of the lamina 


and write M,(a) = mX N,(b) = mY 


we have a point (X, Y) which is called the mass-centre or centre of 
gravity of the distribution. If we take a uniform surface distribu- 
tion, we may disregard the notion of mass, as (X, Y) is then deter- 
mined by the shape of the area only ; the point is then called the 
centroid of the area. 

A point with corresponding geometrical and physical properties 
exists for every solid and every solid mass, but at present we shall 
restrict ourselves to the case of a solid bounded by a surface of 
revolution ; we take a plane of reference perpendicular to the axis 
of revolution and construct a sum by multiplying all the small 
elements of the volume by their distances (x) from the plane of 
reference. Thus we obtain the conception of a first moment of a 
volume with regard to a plane. If a section is made by a plane 
whose coordinate is w, such that the volume included by it and the 


144 INFINITESIMAL CALCULUS CHAP. XI 


plane of reference is V(«), and if we call the first moment of this 
volume with regard to the plane of reference M,(z), then we have 
by the method followed throughout this chapter 
OM,(~) = x x volume of a slice of the solid bounded by the 
planes whose coordinates are w and x + 6a 
= mry*oa 
dM, 


Therefore wi mary” =x f(r) ]* 


where y = f(x) is the equation of the generating curve. 
Now, if the volume is bounded by the planes x = a, x = 5, its first 
moment is M, (b) 
i 


and the z-coordinate of the centroid is 


M,(0)/V 


131. Illustrative example. 
Ex. To find the centroid of a hemisphere. 


Now eu Eee sig ean (Ciro ie 
M,(x) = x(da%x* — 424), since M,(0) = 0 
M,(a) = 3na4 

and X = M,(a)/V = 4rat/gra® = 3a 


EXERCISES XI 


1. Taking m as the mass of a rod OA, whose length is / and whose 
mass-centre is at G, prove that when p= kx 
i. with O as base-point, M, = 3ml, M, = $m? 
ii. with A as base-point, M, = 4ml, M, = imi? 
iii. with G as base-point, M,=0, M, = yumi? 


2. When oe = kz?, prove that 
i. with O as base-point, M, = 3ml, M, = 2mi/? 
ii. with A as base-point, M, = jml, M, = 7yml? 
iii. with G as base-point, M,; = 0, M, = ml? 
3. The thickness at (a, y), a point of a rectangular lamina whose sides 
area = ta,y= + 6,ist(1 — z/a*). Show that m = Stab, M, = zma?. 
4, The shape of a uniform lamina is one-half of a parabolic segment 
cut off from y? = 4ax by « = 2. Show that 
M, = 5ma, N, = gmyy M, = 7mx? N, = my? 
5. A segment of a parabola y? = 4ax, whose furthest corner is (x, y,), 


revolves about Ox. Show that the moment of inertia about the axis 
of the uniform solid paraboloid thus defined is }my,?. 


ANSWERS 


Exercises I. P. 16 


a4a+1 ii, -3+4 iii, Ow -3 iv. Qu “ve 
vi. Ox +2 vi. —“%+7 vill. +6 ix. —5x 
i. ta? Uwe +o iii, —x?+a+1 iv. x27-xa+1 
Ve 2+ 2a vi. 132? — 45x + 30 vii. 0°90? +22-2 viii. 4u?-6x+2 
i. [—o, —1) ii. [—o, 2°5] ili, [1, 0] iv. [-—o, l) 
v, [-«#, -l1] vi. [8, o ] vii. [—0, 1][3, 0] 
viii. (—1, —2) x0 (0. 2°) x. [0, 2°5] xi. [—1, 1°35] 
xli. [—o«, —24)(#, 0] xiii. [-24, 2] xiv. (— 24, 0](%, 0 | 
xv. [—o, —1][0, 1] xvi. [—1, 0] [4, ©] xvii. [—o, O][0, 2] 
xviii. [0, 2] xix. (-2, 7] xx, (—1, 3] xxi. (0, 4] 
xxl. (0, 1] xxiii. [—c, —10][-2,2][4,0] xxiv. [-o, —3](-2, 0](1, o] 
xiv. —2, 1ifl, 2} xxvi, [-2, -1][-1, -75] 
oes) Sy a oteteks. Bs 8s oy 830 
Exercises II. P. 33 ° 
i #=-1], Llm=-a@, Rlim=o i. = —)], no Lilim, Rlim=o 
li, e=-1, Llim=o, Rlim=-o iv. = —1, Lim=Rlim=0 
y. c=—1, noLiim, Rlim=—o; x=1], Lhm=oa, no Rlim 
Vee |. Ulimece . vw lini oo. oa Oo lim = lim=) 
vii. c=-1, Llim=-o, noRlim; 2z=1, no Llim, Rlim=0 
vill. 2=—-—4, Llim=-o, Rlim=o; x=], Llim=Rlim=0 
ix. e=2nr; Llim =-o, Rlim=o; x=(2n+1)r, Llim=o, Rlim=-«a 
x. ©=34(2n4+1)7, Llim=o, Rlim=o 
xl. Defined in [n7, 4(2n+1)7]; w=nr, Rlim= —-0; x=43(2n+1)7, Llim=o 
x. c=n7, Llim=-o, Rlim= ow; x=$(2n+1)7, Llim=o, Rlim=o 
eS ee ery ii F005 = L)(T 00 | lib ne) iv. [—0, 1)(2, 0] 
v. [-2, O](1, o] vi, |= oo, O}[0, 1] [100] vil. (—1, 0](1, 0] 
viii. ... (0, +) (2, 37) ... ix. ...(—Zm, 47) ($27, 427)... x. [0, 0] 
Risa( lt, on] xi. ...[0, 4r][7, $7]... xill, ...[0, 42][47, 7]... 
xiv. [0, 1][1, 0] xv. ... [47, $r][$7, 427]... SViept dee ooo! 
TU Tink liek hie ard vee Ve Oe vi. +1 


Exercises III(A). P. 37 

12 2. 2x 3. 6%+1 4, —2x-3; [-o, 0][0, 0] 

. —(@+1)?; [-«, -1][-1, o] G24 7. 82+4 
enor ai 


ii ANSWERS 


8. —2Qx(x2 ee f=} -1][-1, Uf oo) 9 8 diet lie eee 


10, —(8-2x)-?; [-0, 14] ° 11. —4e(@2e 1) "12, Grrl) sae eee 


Exercises III(B). P. 46 


___ 2a ry _ Aarx rr 2a, eee, 4a*x 
(a-+2)? , (a? ra x2)? . (a— x)? : (2? fa a*)? 
, 2a) ee elpetd boteeribD | Demy 8 cel vis 
(l-a2+<2?)? (l+a” +7)? (l-x)? (1+2)? 
pe? (1) 2(x?-1) . 2 a = (x? +6x+7) 
™ (=a) * + @P a oo) ce CECE) 
eee etal) ga gael lees) ay! Ral eg eee Renae 
(a — a)?(a — b)? (a — 3)?(a@ +2)? (aw + 1)?(a” — 2)? oa 
re a ] 1 : : 
xvi. —1 sag) Pam ed oa Bee ae Fe ix ee : x(2-3sin*x 
XVH XV1ll SEES xix CET: XX. Sitar ( ) 
: -. SIN®—Xcosx ee COS co La sin &, 
Xxi. COSH—UsiIN« 994 1 eee seal eat ee. Stee eee » 4 big ene 
sin?x (1 — sin x)? cos*x 
sin’ — cosa 2 Ls —2 
<V. SSViy 2 XXVil. ——_—_ ee 
sin?x cos?x (sin x + cos x)” (sin « —cos x)? 


Exercises IV. P. 56 


1S) o£, min. ll, @=4, MAX.3 = oe. 
WP evita en ttebyed o4sh— ay rt es iV. 2=3, maxes 2=—4, spin. 
Vv. = —4, min. vi, G=0'31,/3:19, max) ee 
Vi. e= — 2.1, nvin, r= max, Vill. ess Makes tea iin 

ee fle oo, Max. ¢ = 0,min, ll. «= —2, min. in: wey nin, 


3 1. f(2)=0, min. ; (24) =1/27, max. 11. 7(0)=0, f(2)=0, min. ; 7(1 j= sy meee 
iil. £(0)=0, stationary ; f( — ?)= —0°105, min. 
iv. f(-1)=0, stationary; /(0°2)=1°106, max.; /(1)=0, min. 


5. 2° + 6x? - 15x 
G2 74. in ents min. ii. f(4) =0°843, min. 
iii, f(-1)=3, max. ; J(1)~ 3 min. 


iv. J ( Pate —4, min.; /(4(1+,/5))=4, max. 


7. i. e=2, min.; e=6, max. ii, x=$(—/2+,/6),.min.; c=4(./2+,/6), max. 
iii, w= —,/5, max. (—45,/5); x=,/2, max. (4./2); «= —,/2, min, (—4,/2)- 
x=,/5, min. ($5,/5) 
8. i. w=(2n+4)7, max.; x=(2n+ 4), min. 
ii, ©=(2n+4)7, max. x=(2n—-4)z7, min. 
ili. =43(4n—-—1)7, max.; x=4(4n+4+1)7, min. 
iv. ©=$(2n-1)x, nr+(--1)"sin—!(1/,/6), max. 
v=43(2n+1)r, nr —(-1)"sin-!(1/,/6), min. 
v. ©=(2n+1)7, 2nw+cos—1(1/,/3), max. 
x=2nr, (2n+1)r+cos—*(1/,/3), min. 
vi, e=$(4n+1)7r+3a, max.; «=$(4n—1)r+ ha, min. 


ill 


ANSWERS 


Exercises V. P. 72 


1. i. 3(a—1)P-2 ii. 2(% —1)?4+382 ili. 7(x+1)?-14 
iv. 4—(x%+8)? v. («+1)?+2 vi. (w-1)?-4 
4, i. (w?+a4+1)(x?-2+1) 
ii. (v?-ax-1)(x?+a-1), (2?-,/5a+1)(a?+/5e4+1) 
iil. (7?+,/60 +3) (x?-,/6x +43) 
ie ior ii, —4} iii. 4, —2, - 6,5, 
9 i. R(-—2)=14, max.; R(0)=0, min. ii. R(O)=4, max. 
ili. R(-6)=14, max.; R(14)= — 3, min. 
iv. R(14)= — 3, max.; R(-6)=2, min. v. R(-4)=0, min. 
vi. R(-8)=14, max.; R(-2)= —2, min. 
vii. R(-3)=14, max.; R(3)=#, min. 
vill. R(4°4)= —31°5, max.; R(2°1)= —0°5, min. 
ix. None x. R(-1)=14, min. 
xi. R(-b/a)=0, max., R(b/a)=4ab, min., if b/a is positive 
xii. R(-2+.,/3)=2+./3, max.; R(-2-,/3)=2-,/3, min. 
Exercises VI(B). P. 82 
1. 4(2x%+1) 2. 4(2x-1) 3. —9(2-32) 4, -$3,/(2-32) 
5 2(a4-1) esl 6x: (1+ 2x) 
oe 2208, (a2 — 1) (1+3z)° 
8, —- 6x(1 i. 32) 9, be 1 10. Qa 
(1 — 22)? (1+ x)n/(1 — x?) (1 — x),/(1 — x*) 
ie One Ce 13, (1+2)%1—2)(1—5a) 
» (a= 0) (a? + 0)8 
14, -10x(1-%)(2%+3)? 15. (w+a)P—(xt+b)i-{(p+q)x+bp+aq} 
a? a? — 2a*x 
gpa. ele Le sents Tisch sce 1B Re retest peserearort 
es x*/(a? + x?) af x? /(a — x?) (a? + 2%) /(a4 — x4) 
19 2na(a+a)r—1 20 a(a— x) x + 2a%x — a3 
5 (— a)rti 4 J(a? + ac2)3 : nJ (x? Re a*)3 
22, n{(1+2)""!-(l-—2)""}} Bo ie few (1 oan) oa (rae ky 
24. -—ax-?-a®a-2(a?- a2)" 3 25. 2n(b+cx)(a+2bx + cx)! 
- 15 —1+a%?~-2a4 
26. aT de ee Py ie be Eke | 
28a (30+ 7) Pata) PL aa) 
29. 21x7,/(5 + 22) 30. —63(3- 5ar2)2a-8 31. sin 2x 32. 2sin 4a 
33. tcos 4a,/cosec hx 34, -—2sin 2a 35. —sin 2a,/sec 2x 
36. 4sec?2x tan 2x 37. 0. 38. 2sin2x 39. —3sin3z 
40. 9tan?3x sec?3x ao 96-2 42. 3sec a tan?x 


. —2cos x(x sin x+cos x)/x? 


. 4tan 2(%+a) sec?2(%+a) 


—(1+sin x)- 


48. —cosx(4+cosec?x) 


44, 2x(sin x — x cos x) cosec®x 
46. cos «(5-4 cos?a) (4 cos? — 3)-? 
49, 50. 


sin?x sin? 


iv ANSWERS 
51. costx 52. —cot?2 53. tan®a 54. tanta 
55. §a%, puP-q 56. se (ao 57. -1 
58. -—cota, —3cos 3x/2 sin 2x 59. —%secx 60. cosec x, sin 
Exercises VII. P. 89 
Leet Ze -y=3 i. 34+2y+12=0 ill. 4a -y=5 iv. 8y-4x”=5 
v. 82% —39y=168 vi. 8u+y=5 vii. 138” —3y=27 
vill. 2y —,/3%=1-4%,/3r ix. 4u-y=7-1 xX. ©+7yY=T 
4. xjat+y/b=2; x = kai?, y= kor, kn a4 pn | = 23 n even; - 2, four 
normals, n odd, one. 
5. Tangents, y-2x~=a, y=2a; normals, x+2y=2a, x=a 
8. w2=2, $ 18. Qxt-y(1-?t)=at?; y=0, y=a 
Exercises IX. P. 119 
(A constant C may be added to each answer) 
l, 7+4u7+ 423 2. da —ha?-Qx 3. —4(1-a)4 
4, x+a-1+ha-? 5. ata — 2a7x3 + 125 6. «—4a?+423- fat 
7. 4002+ tbat 8. fucu®+4(ad+be)z*+bdx 9 «xPt!/(p+1)+a!-?/(1-p) 
10. $23+2,/x 11. 2,/x(4u+1) 12. -—x-!-4a-?- 42-3 
13. $a? —logx 14. 423+ 447?+4+ log (a - 1) 15. -—4log (1-22) 
16. x-2log(2e+3) 17. log[(a—2)3/(w — 1)2] 18. flog [(2+a)/(2—2)] 
19. 4 log[(x -3)/(x-1)] 20. «+ log (a?-—x2+1) 
21. ta-* log [(x? - a”)/(x?+ a?)] 22, —4(x?+a7)-} 
23. 4x3 —ha2+ax—2log (a +1) 24. log [(2a+1)2(x —1)] 
25. log [(3x —2)/(2%+3)] 26. log [aa —-1)?/(~+1)] 
27. log [a "(a+4)]-4/x 28. 323 29, —4a-? 
80. 4,/(2~+1)° 31. —4(2%4+1)-} 82. (ax+b)"t1/(an +a) 
38. 42° -2a-a7} 34, te? +hatt+aet+ar%t+e 35. a? +z?-a2 
36. 42°+2x%+log x 37. 40°-ha*+x—-log(w@+1) 38. 3(a?-3)%/x 
39. (2u?-4x%4+2),/x 40. log[a—(x—-1)] 41. log [a(a—1)-?] 
42. 2+ log [a?/(a-1)] 43. 2 log (a#°+ 3a - 5) 44, x-log(a?+2”+1) 
45. 4log[(a?-a+1)/(x?+2+41)] 46. x+2logu—-1/x 
47. x-2log(*+1)-(#+1)"! 48. Slog (2x — 3) — 29/(2u - 3) 
49. log [a(x+2)(x+1)-?] 50. log w—2(2x-1)7} 
51. 4 log (x-1) 52. 4log[(a%-1)*/(22+a”+4+1)] 
53. 4 log [a-*(x -1)?/(~+1)] 54, flog [(~+1)(x-1)*]-4(a-1)7} 
55. 4 (a+ 403+ 12a? 4-402) — $ (12a — 11)(*-1)-*?+ 15 log (a - 1) 


56. 
59. 
62. 
65. 
68. 
71. 


73. 


75. 


ANSWERS Vv 


4,/(a? + x?)8 57. ./(a? +2?) 58. (a®+a")Pt1/(np +n) 
4 (log x)? 60. log log a 61. -—4cos3x 

cos 9.0 — 9 cos 8.x 68. x+sin’x 64. log (1+sin 2) 

— log (1 — sin x) 66. tanx-ax 67. dtan?a—-tanx+a 

4 tan?x + log cos x 69. stan?w+tanx 70. 4sec?x — sec x 


4xu+4log(sinw+cosx) 72. [(ac+bd)x+ (be — ad) log (c sinx+d cos x)]/(c? +d?) 


cos (m+n) x } cos (m —n)x 4. 1 sin (m — 7) % ,sin(m+n)x 
mr m+n m—n Benin ae m+n 
rn sin (m — 2) x +3 sin (m + n) 2x 
7 m—nN m+n 


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UNIVERSITY PRESS, GLASGOW, 


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